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COMPUTING MECHANISMS AND LINKAGES BY ANTON IN SVOBODA EDITED BY HUBERT M. JAMES <^s "^ 492 ^2^.22. DOVER BOOKS ON ENGINEERING AND ENGINEERING PHYSICS Supersonic Aerodynamics, E. R. C. Miles. $1.45 Fares Please! A Popular History of Trolleys, Horse-cars, Street-Cars, Buses, Elevateds and Subways, John A. Miller, $1.50 Engineering Mathematics, Kenneth S. Miller. $2.00 Theory of Flight, Richard von Mises. $2.95 The Scientific Basis of Illuminating Engineering, Parry H. Moon. $3.25 Microwave Transmission Design Data, Theodore Moreno. $1.65 Principles of Mechanics Simply Explained, Morton Mott-Smith. $1.00 Heat and Its Workings, Morton Mott-Smith. $1.00 Concepts of Energy Simply Explained, Morton Mott-Smith. $1.25 Methods in Exterior Ballistics, Forest R. Moulton. $1.75 Introduction to Applied Mathematics, Francis D. 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This Dover edition, first published in 1965, is an unabridged republication of the work first published by McGraw-Hill Book Company, Inc., in 1948. It is made available through the kind cooperation of McGraw-Hill Book Company, Inc. This book was originally published as volume 27 in the Massachusetts Institute of Technology Radi- ation Laboratory Series. Library of Congress Catalog Card Number 65-22736 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014 Foreword The tremendous research and development effort that went into the development of radar and related techniques during World War II resulted not only in hundreds of radar sets for military (and some for possible peacetime) use but also in a great body of information and new techniques in the electronics and high-frequency fields. Because this basic material may be of great value to science and engineering, it seemed most important to publish it as soon as security permitted. The Radiation Laboratory of MIT, which operated under the super- vision of the National Defense Research Committee, undertook the great task of preparing these volumes. The work described herein, however, is the collective result of work done at many laboratories, Army, Navy, university, and industrial, both in this country and in England, Canada, and other Dominions. The Radiation Laboratory, once its proposals were approved and finances provided by the Office of Scientific Research and Development, chose Louis N. Ridenour as Editor-in-Chief to lead and direct the entire project. An editorial staff was then selected of those best qualified for this type of task. Finally the authors for the various volumes or chapters or sections were chosen from among those experts who were intimately familiar with the various fields, and who were able and willing to write the summaries of them. This entire staff agreed to remain at work at MIT for six months or more after the work of the Radiation Laboratory was complete. These volumes stand as a monument to this group. These volumes serve as a memorial to the unnamed hundreds and thousands of other scientists, engineers, and others who actually carried on the research, development, and engineering work the results of which are herein described. There were so many involved in this work and they worked so closely together even though often in widely separated labora- tories that it is impossible to name or even to know those who contributed to a particular idea or development. Only certain ones who wrote reports or articles have even been mentioned. But to all those who contributed in any way to this great cooperative development enterprise, both in this country and in England, these volumes are dedicated. L. A. DuBridge. Preface The work on linkage computers described in this volume was carried out under the pressure of war. War gives little opportunity for the advancement of abstract knowledge; all efforts must be concentrated on meeting immediate needs. In developing techniques for the design of linkage computers, the author has therefore been forced to concentrate on finding practical methods for the design of computers rather than on developing a unified and systematic analysis of the subject. The war has thus given to this work a special character that it might not otherwise have had. The impulse to the development of the methods presented in this volume for the mathematical design of linkage computers grew out of a collaboration of the author with his friend, Dr. Vladimir Vand. That col- laboration was begun in France in 1940, and was brought to a premature end by the progress of the war. Though these ideas and methods have largely been developed by the author since that time, he wishes to emphasize that credit for the initiation of the work is shared by Dr. Vand. It must be mentioned also that the techniques described in this book were for the most part developed before the author became asso- ciated with the Radiation Laboratory. The author wishes to express sincere gratitude to Dr. H. M. James, the editor of this volume, who gave the book its present form, contributing many examples and many improvements to the methods. (Sees. : 6-7, 6-8, 645, 8-6.) The book would never have been completed in such a short time with- out the assistance of Miss Constance D. Boyd, who read the manuscripts, and Miss Elizabeth J. Campbell, Mrs. Kathryn G. Fowler, Miss Virginia Driscoll, and Miss Patrica J. Boland, who calculated the tables and drew nomograms. The author also wishes to thank Dr. I. Maddaus, Jr., for bibliographical research. A. Svoboda. Praha, Czechoslovakia, June, 1946. vu Contents FOREWORD by L. A. DuBridge v PREFACE vii Chap. 1. COMPUTING MECHANISMS AND LINKAGES 1 Introduction 1 11. Types of Computing Mechanisms 1 1-2. Survey of the Problem of Computer Design 2 1-3. Organization of the Present Volume 5 Elementary Computing Mechanisms 6 14. Additive Cells 6 1-5. Multipliers 12 1-6. Resolvers 15 1-7. Cams 19 1-8. Integrators 23 Chap. 2. BAR-LINKAGE COMPUTERS 27 2-1. Introduction 27 2-2. Historical Notes 28 2-3. The Problem of Bar-linkage-computer Design 31 24. Characteristics of Bar-linkage Computers 32 2-5. Bar Linkages with One Degree of Freedom 34 2-6. Bar Linkages with Two Degrees of Freedom 37 2-7. Complex Bar-linkage Computers 40 Chap. 3. BASIC CONCEPTS AND TERMINOLOGY. . 43 31. Definitions 43 3-2. Homogeneous Parameters and Variables 47 3-3. An Operator Formalism 49 34. Graphical Representation of Operators 51 3-5. The Square and Square-root Operators 54 Chap. 4. HARMONIC TRANSFORMER LINKAGES 58 The Harmonic Transformer 58 41. Definition and Geometry of the Harmonic Transformer 58 4-2, Mechanization of a Function by a Harmonic Transformer. ... 61 ix CONTENTS 4-3. The Ideal Harmonic Transformer in Homogeneous Parameters . 62 4-4. Tables of Harmonic Transformer Functions 63 4-5. Total Structural Error of a Nonideal Harmonic Transformer . . 67 4-6. Calculation of the Structural Error Function 8Hk of a Nonideal Harmonic Transformer 68 4-7. A Study of the Structural Error Function SH* 71 4-8. A Method for the Design of Nonideal Harmonic Transformers . . 75 Harmonic Transformers in Series. . . . , 77 4-9. Two Ideal Harmonic Transformers in Series 77 4- 10. Mechanization of a Given Function by an Ideal Double Harmonic Transformer 79 4-11. Preliminary Fit to a Monotonic Function 82 4-12. Preliminary Fit to a Nonmonotonic Function 89 4-13. Improvement of the Fit by a Method of Successive Approxima- tions 91 4-14. Nonideal Double Harmonic Transformers 95 4-15. Alternative Method for Double-harmonic-transform ei Design . . 101 Chap. 5. THE THREE-BAR LINKAGE 107 51. Fundamental Equations for the Three-bar Linkage 107 5-2. Classification of Three-bar Linkages 108 5-3. Singular Cases of Three-bar Linkages 112 5-4. The Problem of Designing Three-bar Linkages 117 The Nomographic Method 118 5-5. Analytic Basis of the Nomographic Method 118 5-6. The Nomographic Chart 120 5-7. Calculation of the Function Generated by a Given Three-bar Linkage 122 5-8. Complete Representation of Three-bar-linkage Functions by the Nomogram 125 5-9. Restatement of the Design Problem for the Nomographic Method 127 5-10. Survey of the Nomographic Method ' 128 5-11. Adjustment of b2 and a, for Fixed AXi, AX2, &i 132 5-12. Alternative Methods for Overlay Construction 136 5-13. Choice of Best Value of 6i for Given AXX, AX2 137 5-14. An Example of the Nomographic Method 139 The Geometric Method for Three-bar Linkage Design 145 5-15. Statement of the Problem for the Geometric Method 146 5-16. Solution of a Simplified Problem 147 5- 17. Solution of the Basic Problem 151 5-18. Improvement of the Solution by Successive Approximations. . . 154 5-19. An Application of the Geometric Method: Mechanization of the Logarithmic Function 156 CONTENTS xi Chap. 6. LINKAGE COMBINATIONS WITH ONE DEGREE OF FREE- DOM 166 Combination of Two Harmonic Transformers with a Three-bar Link- age 166 6-1. Statement of the Problem. . . : 166 6-2. Factorization of the Given Function 168 6-3. Example: Factoring the Given Function 171 6-4. Example: Design of the Three-bar-linkage Component 174 6-5. Redesign of the Terminal Harmonic Transformers 186 6-6. Example: Redesign of the Terminal Harmonic Transformers . . . 187 6-7. Example : Assembly of the Linkage Combination 193 Three-bar Linkages in Series 195 6-8. The Double Three-bar Linkage 195 Chap. 7. FINAL ADJUSTMENT OF LINKAGE CONSTANTS 199 7-1. Roles of Graphical and Numerical Methods in Linkage Design . . 199 7-2. Gauging Parameters 202 7-3. Use of the Gauging Parameter in Adjusting Linkage Constants . . 201 7-4. Small Variations of Dimensional Constants 205 7-5. Large Variations of Dimensional Constants 205 7-6. Method of Least Squares 206 7-7. Application of the Gauging-parameter Method to the Three-bar Linkage 207 7-8. Application of the Gauging-parameter Method to the Three-bar Linkage. An Example 209 7-9. The Eccentric Linkage as a Corrective Device 217 Chap. 8. LINKAGES WITH TWO DEGREES OF FREEDOM 223 8-1. Analysis of the Design Problem 223 8-2. Possible Grid Generators for a Given Function 226 8-3. The Concept of Grid Structure 228 8-4. Topological Transformation of Grid Structures 232 8-5. The Significance of Ideal Grid Structure 233 8-6. Choice of a Nonideal Grid Generator 238 8-7. Use of Grid Structures in Linkage Design 243 Chap. 9. BAR-LINKAGE MULTIPLIERS 250 9-1. The Star Grid Generator 250 9-2. A Method for the Design of Star Grid Generators with Almost Ideal Grid Structure 251 9-3. Grid Generators for Multiplication 256 9-4. A Topological Transformation of the Grid Structure of a Divider . 258 9-5. Improvement of the Star Grid Generator for Multiplication . . . 264 9-6. Design of Transformer Linkages 271 9-7. Analytic Adjustment of Linkage Multiplier Constants 277 9-8. Alternative Method for Gauging the Error of a Grid Generator . 281 xii CONTENTS Chap. 10. BAR-LINKAGE FUNCTION GENERATORS WITH TWO DE- GREES OF FREEDOM 284 10.1. Summary of the Design Procedure 284 10-2. Example: First Approximate Mechanization of the Ballistic Function in Vacuum 286 10-3. Example: Improving the Mechanization of the Ballistic Function in Vacuum 292 10-4. Curve Tracing and Transformer Linkages for Noncircular Scales 295 APPENDIX A. Tables of Harmonic Transformer Functions 301 APPENDIX B. Properties of the Three-bar-linkage Nomogram . . . 333 INDEX 353 CHAPTER 1 INTRODUCTION l'l. Types of Computing Mechanisms. — Computing mechanisms may be divided into two distinct types : arithmetical computing machines, familiar to the layman through their common use in business offices, and continuously acting computing mechanisms and linkages that range in complexity from simple cams and levers to enormously complex devices for the direction of naval and antiaircraft gunfire. The arithmetical computing machines accept inputs in numerical form, usually on a keyboard, and with these numbers perform the simple arithmetical operations of addition, subtraction, multiplication, and division — usually by the iteration of addition and subtraction in counting devices. The results are finally presented to the operator, again in numer- ical form. In their simplest forms these machines have the virtue of applicability in a wide variety of computations, including those requiring very high accuracy. By elaboration of these devices, as by the introduc- tion of punched-tape control, their possibilities for automatic operation can be greatly increased. Characteristic of their operation, however, is their production of numerical results by calculations in discrete steps, involving delays which are always appreciable and may be very large if the required calculation is of complex form. Continuously acting computing mechanisms are less flexible and have less potential accuracy, but their applicability to the instantaneous or to the continuous solution of specific problems — even quite complex ones — makes them of great practical importance. They may serve as mere indicators of the solutions of a problem, and require further action by human agency for the completion of their function (speedometer, slide rule) ; or they may themselves produce a mechanical action functionally related to other mechanical actions (mechanical governors, automatic gunsight) . Continuously acting computers fall into two main classes: function generators and differential-equation solvers. Function generators pro- duce mechanical actions — usually displacements or shaft rotations — that are definite functions of many independent variables, themselves intro- duced into the mechanism as mechanical actions. Simple examples of such mechanisms are gear differentials, two- and three-dimensional cams, slide multipliers and dividers, linkage computers, and mechanized nomo- grams. Computers of the second class generate solutions of some definite 1 2 INTRODUCTION [Sec. 12 differential or integrodifferential equation — often an equation that involves functions continuously determined by variable external cir- cumstances. Elementary devices of this type are the integrators, com- ponent solvers, speedometers, and planimeters. From these elementary devices one can build up complicated mecha- nisms that perform elaborate calculations. We may mention their application in gunsights, bombsights, automatic pilots (for airplanes, submarines, ships, and torpedoes), compensators for gyroscopic com- passes, tide predictors, and other robots of varied types. The present volume will deal only with the problem of designing con- tinuously acting computing mechanisms. 1*2. Survey of the Problem of Computer Design. — There is no set rule or law for the guidance of a designer of complex mechanical com- puters. He must "weigh against each other many diverse factors in the problem: the accuracy required; the cost, weight, volume, and shape of the computer; its inertia and delay in action; the forces required to operate it; its resistance to shock, wear, and changes in weather conditions. He must consider how long it will take to design the computer, how easily it can be built, how easily it can be operated by a crew, whether suitable sources of power will be available, and so on. The complexity of the theoretical and practical problems is so great that two designers working on a given problem will never arrive at precisely the same solution. For practical reasons, a designer should be asked to find a computer that meets certain specified tolerances, rather than the best possible computer for a given use. He should know what will be the maximum tolerated error of the computer, the maximum cost, weight, and volume occupied, the maximum number of operators in the crew, the maximum number of servomechanisms allowed, and so on. Tolerances provide a convenient means for controlling the development of the computer, and — if established in a practical way — they permit some freedom of choice by the designer. Choice of Approach to the Design Problem. — The type of computer to be built is sometimes indicated in the specifications. If not, the first task of the designer is to decide whether the computer is to be mechanical, electrical, optical, or a combination of these. At the same time that this important decision is made, the designer must weigh in his mind the path that his thinking will follow. There are two principal methods for design- ing a computer: the constructive method and the analytic. The constructive method makes use of a small-scale model of the real system with which the computer is to deal. For example, a constructive antiaircraft fire-control computer might determine the elements of the lead triangle by maintaining within itself and measuring the elements of a small model of this triangle. Sec. 1-2] SURVEY OF THE PROBLEM OF COMPUTER DESIGN 3 In using the analytic method, the designer concentrates on the analytic relations between the variables involved. A relation between variables, such as Z = Xy + ?, (1) can be given mechanical expression in terms of displacements or shaft rotations, without regard to the nature of the quantity represented by the variables x, y, and z. For example, one may possess two devices that generate output displacements xy and x/y, respectively, given input dis- placements x and y. Combining these with a third device for adding their output displacements, one can then produce a computer that, given input displacements x and y, generates a final output displacement z having continuously the value specified by Eq. (1). The computer is then a " mechanization' ' of Eq. (1), rather than a model of any special system involving variables x, y, and z thus related. Computers designed by analytic methods consist of units ("cells") that mechanize fairly simple relations, so connected as to provide a mechanization of a more complex equation or system of equations. For any given problem a great variety of designs is possible. This variety arises in part from the possible choice among mechanical cells mechaniz- ing a given elementary relation, and in part from the variety of ways in which the relation between a given set of variables can be given analytic expression. Thus, each of the equations z = I (y> + 1), (2a) = *(v + l)> (26) zy = x(y2 + 1), (2c) [all equivalent to Eq. (1)] suggests a different method of connecting mechanical cells into a complete computer. This flexibility in analytic design methods makes it possible to arrive at designs that are in general more satisfactory mechanically than those obtained by constructive methods. In the present volume we shall be concerned entirely with mechanical computers designed by the analytic method. Block Diagram of the Computer. — To each formulation of the problem in analytic terms there corresponds a block diagram of the computer. In this diagram each analytic relation between variables is represented by a square or similar symbol, from which emerge lines representing the variables involved ; a line representing a variable common to two relations will connect the corresponding squares in the diagram. In mechanical terms, each square then represents an elementary computer that estab- 4 INTRODUCTION [Sec. 1-2 lishes a specified relation between the variables, and the connecting lines represent the necessary connections between these elementary com- puters. By examination of block diagrams the designer will be able to see the principal virtues of each computing scheme : the complexity of the system, the working range of variables, the accuracy required of individual components, and so on. On this basis he can make at least a tentative selection of the block diagram to be used. Selection of Components for the Computer. — Knowing the accuracy and mechanical properties required of each computing element, the designer can select the elementary computers from which the complete device is to be built. As an example of the diverse factors to be borne in mind, let us suppose that it is required to provide a mechanical motion proportional to the product of two variables, X\ and X2. A slide multiplier of average size will allow an error of from 0.1 per cent to 0.5 per cent of the whole range of the variable ; this error will depend on the quality of the construction — on the backlash and the elasticity of the system. A linkage multiplier will have an error of some 0.3 per cent due to its structure, practically no error from backlash, and a slight error due to elasticity of the system if the unit is well designed; the space required by a linkage multiplier is small, but its error cannot be reduced by increasing its size. If these devices do not promise sufficient accuracy, the designer must use multipliers based on other principles. It is possible to perform multiplication by use of two of the precision squaring devices illustrated in Fig. 1-23, by connect- ing these in the way suggested by the equation XrX2 = i(Xj + x2y - i(Xi - x,)\ (3) The error of such a multiplier may be as low as 0.01 per cent, but the system has an appreciable inertia. About the same accuracy is attain- able by a multiplier based on the differential formula for multiplication, d(XiX2) = Xi dX2 + X2 dXi; (4) this employs two integrators, and is commonly used when two quantities are to be multiplied in a differential analyzer. This scheme is useful only when it is possible to allow a slow change in a constant added to the product XiX2 — a change which will result from slippage in the integra- tors, negligible for a single multiplication but accumulating with repetition of the operation. From this discussion it should be evident that there is no "best" multiplier. Similarly, other components of a computer must be selected with due regard for their special characteristics and the demands to be made upon them. Mathematical Design of the System. — From the block diagram one should proceed to the mechanical design of a system through an inter- Sec. 1-3] ORGANIZATION OF THE PRESENT VOLUME 5 mediate step — that of establishing the " mathematical design" of the system. The mathematical design ignores the dimensions not essential to the nature of the computation to be carried out — diameters of shafts, dimensions of ball bearings, dimensions of the frame — but specifies the dimensions of levers measured between pivots and joints, the size of fric- tion wheels, tentative gear diameters and gear ratios. The properties of this design should be studied carefully, because this usually leads to a change in some detail of the design, and sometimes even to choice of a new block diagram. Final Steps in the Design. — From the mathematical design of the system one can proceed to the design of a working model. The elements of this model should be accessible rather than massed together, inexpen- sive, and quick to manufacture. If the performance of the working model is found to be satisfactory, the first model can be designed. Here the ingenuity of the designer must be used to the maximum. The parts of the mechanism must be arranged compactly to decrease space require- ments, weight, and the effects of elasticity and thermal expansion, but they should not be massed in such a way that assembly is difficult, or repair or servicing impossible. Sometimes division of the whole computer into several independent parts is advisable. Finally, the computer can be built and tested against specifications. 1«3. Organization of the Present Volume. — It is not possible to dis- cuss in one volume all elements of the problem of computer design. This book will deal principally with bar-linkage computers — specifically, with the mathematical design of elements for such computers. Bar linkages are mechanically very satisfactory, and computers built from them have many important virtues, but the mathematical design of these systems is relatively difficult and is not widely understood. There are few stand- ard bar-linkage elements for computers; it is usually necessary to design the components of the computer, and not merely to organize standard elements into a complex assembly. It is hoped that the design methods to be described here will lead to their more general use. Bar linkages can be used in combination with the standard computing mechanisms. For this reason, and for the contrast with the bar link- ages which are to be discussed later, this volume begins with a brief survey of some more or less standard elements of mechanical computers. Chap- ter 2 is devoted to a general discussion of bar linkages. Chapter 3 establishes terminology and describes graphical procedures of which extensive use will be made. Chapters 4, 5, and 6 discuss, in order of their increasing complexity, bar linkages with one degree of freedom — gener- ators of functions of one independent variable. Chapter 7 indicates some mathematical methods of importance in bar-linkage design. Finally, Chaps. 8, 9, and 10 develop methods for the design of bar-linkage gener- 6 INTRODUCTION [Sec. 1-4 ators of functions of two independent variables — a field in which bar linkages have very striking advantages. ELEMENTARY COMPUTING MECHANISMS The remainder of this chapter will give a brief survey of elementary computing mechanisms, or " cells," of more or less standard type. Dis- cussion of bar-linkage cells will be deferred to Chap. 2. 1*4. Additive Cells. — " Additive" or "linear" cells establish linear relations between mechanical motions of the cell, usually shaft rotations or slide displacements. If these are described by parameters Xi, X2, X3, the cell will compute X3 = Q • X, + Q' ' X2 + C. (5) Here Q, Q', and C are constants depending on the design of the cell and the choice of the zero positions from which Xi, X2, and X3 are measured. By Fig. 1-1. — Bevel-gear differential. proper choice of the zero positions, C can always be made to vanish; in what follows it will be assumed that this has been done. The bevel-gear differential (Fig. 1-1) is a well-known linear cell for which all three parameters are rotations. The parameter X\ is the rota- tion of the shaft Si from a predetermined zero position, Xi = 0; the posi- tive direction of rotation is indicated by symbols representing the head and tail of an arrow with this direction. The parameter X2 is the rotation of the shaft S2 from a similar zero position ; X3 is the rotation from its zero position of the cage C carrying the planetary bevel gears G. The zero positions are not indicated in the figure. The equation of the bevel-gear differential is X3 = 0.5Xi + 0.5X2. (6) To derive this it is convenient to consider the value of X2 corresponding to given values of Xi and X3. Let us consider the differential to be originally in the position Xi = X2 = X3 = 0. The parameters Xi and X3 can then be given their assigned values in two steps, the first a rotation of both the Sec. 14] ADDITIVE CELLS shaft Si and the cage C through the angle X3, and the second a rotation of the shaft Si through an additional angle Xi — X3. In the first step the differential moves as a unit; the shaft S2 is rotated through the angle X3. In the second step, the cage is stationary and the movement of the shaft Si is transmitted to the shaft S2 with its sense of rotation reversed; the rotation through angle Xi — X3 of the shaft Si causes rotation through Xz — Xi of the shaft St. The total rotation of the shaft S2 is then X2 = X3 + (X3 — Xi), from which Eq. (2) follows immediately. It is, of course, essential that all rotations be taken as positive in the same sense. It is remarkable that Eq. (6) is independent of the ratio of the bevel gearing of the differential; the essential characteristic of this type of Fig. 1-2. — Cylindrical-gear differential. differential is that the gearing of the cage transmits the relative motion of the shaft Si to the shaft S2 in the ratio 1 to 1, but with reversed sense. It is not necessary to use bevel gears in the cage to obtain this result; cylindrical gears can accomplish the same purpose. A cylindrical-gear differential is shown in Fig. 1-2. This differential is equivalent to the common bevel-gear differential, except in its mechanical features. It is flatter, and easier to construct in large numbers, but there is one more gear mesh than in the common type; there may be more backlash and more friction. It should be noted, however, that bevel gears are subject to axial as well as radial forces in their bearings, and that these may also increase friction. The spur-gear differential shown in Fig. 1-3 has only two gear meshes, and is quite flat. The planetary gears G in their cage C do not invert the motion of the shaft Si when transmitting it to the shaft S3, but can be made to transmit it at a ratio different from 1. The equation of this INTRODUCTION [Sec. 14 differential is X3 = QX, + (1 - Q)X2. (7) To prove this relation we can use the same method as before. Let us begin by considering the differential in the zero position, Xi = X2 = X3 = U. We wish to find the value of X3 corresponding to given Xi and X2. We introduce the angles Xi and X2 in two steps, first turning both the shaft Fig. 1-3. — Spur-gear differential. x2 Fig. 1-4. — Differential with axially displaced spiral gear. aSi and the cage C through the angle X2, and then the shaft Si through an additional Xi — X2. In the first step the differential is turned as a rigid body; the shaft S2 is also turned through the angle X2. In the second step the shaft S3 is turned through Q(Xi — X2); its total motion is X3 = X2 + Q(Xi — X2), in agreement with Eq. (7). If we make Q = Q' = 0.5 by proper choice of the gear ratios, we can obtain a differential equivalent to the bevel-gear differential. The fact that the free choice of Q gives to this differential a larger field of applica- bility does not necessarily mean that this differential should be preferred Sec. 14] ADDITIVE CELLS 9 to those with Q = 0.5; it is convenient to use differentials with Q = 0.5 as prefabricated standard elements. A differential with axially displaced spiral gear is shown in Fig. 1-4. The parameter X2, which measures the axial displacement of the spiral gear and the pin P2, is variable only within finite limits. The mechanical structure of this differential is, however, much simpler than that of the differentials already mentioned, for which all parameters can change with- out limitation. The equation of this differential is X3 v 2irn v (8) where n is the number of threads per inch along the axis of the spiral gear on the shaft £2 and m is the number of teeth on the gear with which it Differential worm gearing. meshes. The helical angle of the gears should be at least 45° for smooth action and small backlash. The differential worm gearing shown in Fig. 1-5 is used for the same purpose as the preceding differential, especially if the range of values of X2 corresponds to a large fraction of a revolution of the shaft Si or even to several revolutions of this shaft. The equation of this differential is X3 = ± — Xi + -„ X2 (radians) t K (9) where t is the number of teeth of the worm gear, m is the multiplicity of the threads of the worm, and R is the radius of the worm gear. The sign in Eqs. (8) and (9) depends on the sense of the threads of the spiral or worm gear. The screw differential shown in Fig. 1-6 combines an axial translation Xi of a screw with a translation X2 of the nut N with respect to the screw; X3 = Xi + X; (10) 10 INTRODUCTION [Sec. 1-4 To obtain the first translation, the pin P on which the screw turns is displaced by Xi. The rotation of the screw comes from the gear G, which meshes with a cylindrical rack C and slides along it. The real input Fig. 1-6. — Screw differential. parameter of the differential is not X2, but the angle X4 through which the rack is turned. The equation of the differential is then X3 = Xi + kXA. (11) The sign depends on the sense of the screw; A* is a constant determined by the gear ratio, the number of threads per inch on the screw, and their multiplicity. All three parameters of this differential have constructive limits. The belt differential (Fig. 1-7) makes use of the inextensibility of a belting on several pulleys. In practice, chains, strings, and special cables are used as belts. The equation of the belt differential is X3 = C - 0.5Xi - 0.5X- (12) Fig. 1-7.— Belt differential. where C is a constant depending on the choice of zero points of the parameters. The tension in the belt must not fall below zero at any time; if it does, the belt will sag and the equation of the differential will not hold. To obtain positive action in the direction of increasing Xz, it is necessary to preload the belt by putting a load on the output pulley — for instance, by a spring that can exert a force large enough to produce the desired action. The maximum driving force required for this differ- ential will then be about twice the force necessary to operate it without preloading. The loop-belt differential (Fig. T8) has the belting in the form of a loop with length independent of the position of the pulleys. The belt can then Sec- 1-4] MULTIPLIERS 11 be preloaded (turnbuckle B) without adding to the driving force of the differential, except by the increased friction in the bearings. Belt differentials are some- times used to add a large number of parameters; they are easily combined in batteries, as indicated schematically in Fig. 1 -9. In such an arrangement the parameter X7 may have so large a range that it is impractical to use a slide as the output terminal. It is better practice to use a drum (dashed Fig. 1-8. — Loop-belt differential. line in Fig. wound off. 1-9) on which the belt is wound on and at the same time To prevent slippage, the belt should make many turns on the drum and be fastened to it; a chain on chain sprockets may also be used as the belt. The above enumeration does not exhaust the possibilities for linear mechanical cells; there are many variants the use of which may be dictated by special circumstances. As a rule, when a differential is used in a computing mechanism, two ?1 0*) O ^7 °^ ^s members (the input terminals) are moved by external forces; this results in movement of a third mem- ber (the output terminal) which is in turn required to furnish an appreci- able force. If differentials were fric- tionless, any two of their three terminals could be used as input terminals. In reality, only a few of the differentials described here have complete interchangeability of the terminals. For instance, with the screw differential (Fig. 1-6) it is im- possible to have X* as the output parameter if the helical angle of the screw is so low that self -locking of the nut on the screw occurs; it is possible to use X\ as an output parameter, and, of course, also X3. With the differential worm gearing of Fig. 1-5, Xi is an impracticable output parameter. 77777777777777777777777777 Fig. 1-9. — Loop-belt differential for the evaluation of X7 = C - Xi 2X2 + 2X3 - 2X4 + 2X& - 2X6. 12 INTRODUCTION [Sec. 1-5 1»6. Multipliers. — Multipliers are computers that establish between three parameters a relation H-X.Z — Jii ' X.2, (13) where R is a constant that depends on the type of multiplier and on its dimensions. The action of the slide multiplier shown in Fig. 1-10 is based on the proportionality of the sides of two similar triangles. These are triangles with horizontal bases, and vertices at the central pin shown in the figure : Fig. 1-10. — Slide multiplier. the first has a base of length R and altitude Xi, the second a base of length X2 and altitude X3. Thus or R _ X% Xi Xz RXz = X\X%. (14a) (146) The figure gives a schematic rather than a practical design; the lengths of the sliding surfaces as shown are not great enough to prevent self-locking in all possible positions of the mechanism. These lengths determine the space requirements for multipliers of this type; they must be relatively large in two directions. It is difficult to make this type of multiplier precise. The pins in slots, as shown in the figure, are mechanically inadequate, and roller slides on rails must be used. One can not achieve the same end by increasing the dimensions of the multiplier because the Sec. 1-5] MULTIPLIERS 13 elasticity of parts comes into play, not only when the parts are operating in a computer, but also when they are being machined. The slide multiplier shown in Fig. 1-11 saves space in one direction. There are fewer sliding contacts, and the slides are easier to construct. RX^X^Xz Fig. I'll. — Slide multiplier with inputs Xi, Xi — X2. Xl-Olc Xi=Ct -«-2 Fig. 1*12. — Intersection nomogram for multiplication Xi = x/ • xu. This device cannot multiply Xi and X2 directly to compute RXZ = XiX2; the input terminals must be given translations of Xi and Xi — X2. The difference is easy to obtain if the parameters are generated as shaft revolutions before entering the multiplier ; screws can then be used instead 14 INTRODUCTION [Sec. 1-5 of the slides shown in the figure, and the required difference can be formed by a gear differential. Nomographic Multipliers. — A multiplier that is structurally related to a nomogram for multiplication will be called a "nomographic multiplier." Such multipliers can be derived from intersection or alignment nomograms ; the examples to be given here are related to intersection nomograms. Fig. 1-13. — An intersection nomogram for multiplication, obtained from the nomogram in Fig. 1-12 by a projective transformation. Figure 1*12 shows an intersection nomogram for multiplication in an unusual form, the full significance of which will be made clear in the latter part of this book. This represents the formula Xi = XjXk. (15) It consists of three families of lines, of constant Xi, Xj, and Xk, respectively ; through each point of the nomogram passes a line of each family, cor- responding to values of xi} xif and xk which satisfy Eq. (15). (The lines in this particular figure are drawn for values of the x's that are powers of 1.25; this is not of immediate importance for our discussion.) The multi- Sec. 1-6] RESOLVE RS 15 plier of Fig. 1-10 is structurally related to this nomogram. The rotating slide can be brought to positions corresponding to the radial lines in the nomogram; the horizontal and vertical slots correspond structurally to the horizontal and vertical lines on the nomogram, and the pin that con- nects all slides mechanically assures a triple intersection of these lines. The values of xiy x}; and xk corresponding to the positions of the three slides must then satisfy Eq. (15) ; to complete the multiplier it is only necessary to provide scales from which these values can be read, or, as is done in Fig. 1-10, to provide mechanical connections such that terminal displacements are proportional to these quantities. By a projective transformation of the nomogram in Fig. 1*12 one can obtain the nomogram in Fig. 1-13, where lines of constant values of the Fig. 1-14. — Nomographic multiplier. variables Xi, x}-, and xk form three families of radial lines intersecting in three centers. The obvious mechanical analogue of this nomogram for multiplication is shown in Fig. 1-14. It consists of three slides that rotate about centers corresponding to the centers of the radial lines in Fig. 1-13; these slides are bound together by a pin, which establishes the triple intersections found in the nomogram, and the corresponding values of Xi, Xj, and xk are read on circular scales. It will be noted that the scale divisions are not uniform. Such nonuniform scales are of more general use than one might expect. Often one will have to deal with variables generated with nonuniform scales by some other computer; by proper choice of the projective transformation one can then hope to produce a multiplier of this type with similarly deformed scales. 1-6. Resolvers. — The resolver is a special type of multiplier. It generates a parameter X*, and usually also another parameter X*, as a product of a parameter X\ and a trigonometric function — the sine or 16 INTRODUCTION [Sec. 1-6 cosine — of a parameter X2. The equations are X3 = Xi sin X2, (16a) X4 = Xi cos X2. (166) The name of this device is derived from its action as a resolver of a vector displacement into its rectangular components. A simplified design of a resolver is shown in Fig. 1*15. In the plan view, Fig. l-15a, we see the materialization of a vector by a screw: the axis of the screw points in the direction of the vector, at an angle X2 to a zero line; the length Xi of the vector is established as the distance from the pivot 0 on which the whole screw is rotated to a pin T on the nut of the screw. To obtain the components of the vector, slides are sometimes used, as in the case of the multiplier in Fig. 1-10. In Fig. 1-15 there is sug- gested a solution that gives much better precision and saves space. Perpendicular shafts pass through the block B that carries the pin P. These shafts are carried by rollers on rails; their parallelism to given lines is well assured by gears that mesh with racks fastened to the frame. For convenience of construction the axes of the shafts do not intersect with each other and with the axis of the pin T. This introduces a con- stant term e into the displacement of the shafts — that is, it causes a dis- placement e in the effective zero positions of X3 and X4. It is of interest to note how the parameter Xi is controlled from the input shaft 8\ (Fig. 1-156.). While the screw is rotated through the angle X2 on the shaft &2, it is necessary to control the value of Xi by a gear G that rotates freely on this shaft. If such a gear is turned through an angle proportional to Xi — is held fixed when Xi is constant — the screw will spin on its axis whenever X2 is changed ; the length of the vector will be affected by change in X2, and will not represent the desired value of Xi. It is thus necessary to keep the screw without spin with respect to $2 when only X2 is changed — to keep the gear G moving along with the shaft #2 whenever Xi is fixed. This is accomplished by the so-called "com- pensating differential," D. As is shown in the figure, the planetary gear of this bevel-gear differential is geared to the shaft S2 in the ratio 1 to 1; the differential thus receives an input — X2. When the input shaft Se is rotated through X6, the output shaft £5 is rotated through an angle Xh = -X6 - 2X2. (17) By gearing the gear G to the shaft S$ in the ratio 2 to 1, the angle turned by G can be made to be Xo = -0.5X6 = 0.5X6 + X2. Then if S6 is stationary, XQ changes equally with X2, and the screw is not spun; Xi remains constant. If the shaft S& is turned, the gear G turns Sec. 1-6] RESOLVERS 17 with respect to the shaft S2 through an equal angle. The change in Xi is then proportional to the rotation of the shaft S&:Xi — QXh the constant Q depending on gear ratios and the threading of the screw. • X3+e QX^Xs Fig. 1*15. — Resolver. (a) Plan view, (b) Elevation. The teeth of the racks are omitted from the figures. The design in Fig. 1-15 is so oversimplified that the resolver is sure to be lacking in precision. In particular, the flexibility of the structure sup- porting the screw is excessive: shaft S% is easily bent and easily twisted. 18 INTRODUCTION [Sec. 1-6 This can be remedied by placing the screw subassembly on a circular plate with a large ball bearing on its circumference, and using a driving shaft of reasonable diameter. A better construction (but one that is not always usable) is presented in Fig. 1-16. In the plan view, Fig. l-16a, we observe the main difference between the subassembly of the screw in Fig. 1-15 and the present design. (&) ^^ v7/////AV///////////>\ VZZZZZZZZZZZZZZZZZZZZZZZZZ2& Fig. 1-16. — Alternative resolver design, (a) Plan view. (6) Elevation. In Fig. 116a the pin T is carried on an arm of radius R that rotates on a pivot P. This pivot is placed at a distance R from the center S of the circular plate H to which it is fastened. By rotating the arm PT, the vector ST can be changed in length. Its direction would be changed at the same time if it were not for a compensating rotation of the plate H. Since the triangle SPT is isosceles the angle of rotation of ST to be com- Sec. 1.7] CAMS 19 pensated for is exactly half of the angle of rotation of the arm PT. To introduce this compensation a differential is used; to change the direction of the vector ST in a desired manner, the table H is rotated through a second differential. The two differentials, Di and D2, are shown in Fig. 1-166. Their function may be understood in this way. To change the direction of the vector ST we must rotate the whole subassembly of the plate if as a unit; we must turn the gears G\ and G2 by the same amounts. These gears are geared to the cages of the planetary gears of the differen- tials Z>i, D2, at the same ratio (1 to 1 in the figure) ; these also must be turned equally. That is accomplished by turning the shaft S2 and by keeping the shaft Si stationary. To change the length of the vector ST without turning it we have to turn the arm clockwise, for example, in the plan view, and the plate H counterclockwise by half the amount. This is accomplished by turning the shaft Si. This shaft is geared to the input of the differential Di at the ratio 1 to 1 and to the input of the differential D2 at the ratio 3 to 1 ; when the shaft Si rotates, the gear Gi turns three times faster than the gear G2. To see that this gives a compensating rotation of the plate through an angle —X when the arm PT rotates through 2X relative to the plate H, we observe that if the gear G\ were fixed, a rotation of G2 and the plate H through — X would rotate the arm with respect to the plate also by —X. To bring it to the correct posi- tion, +2X, it must then be rotated through an angle of +3X with respect to the plate. To accomplish this the gear Gi must be rotated through an angle — 3X, since the direction of rotation is reversed in the gear (r3. Thus Gi must turn in the same direction as G2, but three times as fast. 1*7. Cams. — A cam is a mechanism that establishes a functional rela- tion between parameters Xi and X2: X2 = F(Xi). (18) If Xi is the input parameter, X2 the output parameter, it is necessary in practice that F(Xi) be a single-valued, continuous function with deriva- tives which do not exceed certain limits. Plane cams exist in two principal variants, shown in Figs. 1-17 and 1-18. In the first the cam has the form of a disk shaped along a general curve. Contact with this cam is made by a roller on an arm ; the contact is assured by tension of a spring. A cam of this type is easy to build and has negligible backlash, but the force on the arm is rather small in one of the two senses of motion — not larger than the force of the spring. In the second variant there is a slot milled into a flat surface rotating on a pivot; contact is made by a roller carried on a slide, as shown in Fig. 1*18. The second form does not permit use of as steep a spiral as does the first, since self-locking is more likely to occur. 20 INTRODUCTION [Sec. 1-7 The cylindrical cam shown in Fig. 1-19 has a slot milled into the surface of a cylinder; a small roller carried by a slide passes along the slot when the cam is turned on its axis through the input angle X\. The form of the slot is so chosen that the motion of the slide, described by the output parameter X2, has the desired character. Fig. 1*17. — Plane cam with spring contact. Fig. 1-18. — Plane cam with groove contact. One variant of pin gearing, as shown in Fig. 1-20, has a gear with a special type of tooth meshing with a milled curved rack. (The milling tool has a cutting shape identical with the shape of the teeth of the gear.) Another form of pin gear (Fig. 1-21) has pins of special shape inserted in a plate; these mesh with a specially formed gear. In both variants the gear is keyed on a shaft, with freedom for lateral motion; this motion Fig. 1-19. — Cylindrical cam with groove contact. Fig. 1-20. — Pin gearing with pins on gear. of the gear is assured by the action of the curved rack on the pins on the gear, or by a special cam constructed for this purpose. The belt cam shown in Fig. 1-22 is a noncircular pulley or drum on which is wound a belt, or string, or some other kind of belting. If the number of revolutions of such a cam is to be greater than one, the string is wound in a spiral; the shape of this spiral should assure a smooth tan- gential winding of the string on the cams. Cams of this type can allow Sec. 1-7] CAMS 21 very large travels of the belt and shaft, but they are mechanically less desirable than pin gears. They are not so safe in operation, and rather Fig. 1-21. — Pin-gearing with pins on the disk. delicate, especially in the compensated form in. which equal lengths of string are simultaneously wound off and wound on. An example of a compensated belt cam is the squaring cam shown in Fig. 1-23. In this, two strings are wound partly on a cylinder, partly on a cone. The winding on the cone is in the form of a spiral with equally spaced threads; the form of the wind- ing is assured by a groove. One string begins on the left side of the drum and, after a number of turns, passes on to the cone and continues in the groove to the tapered right end of the cone. The second string begins on the right side of the cylinder and after several turns to the left passes also onto the cone, where it continues through the groove to the left, to end at the larger end of the cone. The element of rotation dXi of the cone produces a motion of the string equal to RidXh where Ri is the average radius of the cone at the points where the string meets and leaves the cone. The corresponding rotation of the drum is therefore dX2 = (— RidXi)/Rv, Fig. 1-22— Belt cam. 22 INTRODUCTION [Sec. 1-7 where R2 is the radius of the drum. The radius R\ is proportional to the angle Xi measured from a properly chosen zero position of the shaft £1. (This zero position is, of course, not practically attainable, since it would correspond to zero radius of the cone at the point of contact.) We have then kXxdXt — dXo = R< X*~ ~W2Xl (19a) (196) if the zero point for X2 is properly chosen. Here k is the increment of the radius R\ per radian rotation of the shaft S\. Fig. 1-23. — Compensated squaring cam. This squaring cam does not by itself operate down to Xi = 0. It can, however, be used in a range including zero if it is combined with a differ- ential. With Xl = X3 + C, (20) Eq. (19) becomes X2 = KX\ + 2KCX3 + KC\ (21) Introducing the new parameter X4 = X2 - 2KCXl + KC\ we have X4 = KXl (22) (23) this holds even if X3 is zero or negative. The larger the negative values of X3 to be reached the larger must be the positive constant C. The Sec. 1-8] INTEGRATORS 23 precision of a cam of this type can be made very high; the error may be less than 0.02 per cent of the total travel of the output shaft S2. The rela- tively great inertia and bulk of the device (especially when it is combined with a differential for squaring negative numbers), limits its use to cases where precision is essential. Three-dimensional cams or "camoids," such as that shown in Fig. 1-24, are bodies of general form with two degrees of freedom — for instance, a translation of Xi and a rotation X2 — in contact with another body with one degree of freedom, for instance, a translation Xz. The parameter X3 will then be a function of two independent parameters, X\ and X2: Xz = F(X\, X2). (24) The body in contact is called the "follower"; it may be a ball on a slide, as shown in the figure, or an arm rotating on a pin parallel to the main axis of the camoid and touching the surface of the cam. Camoids are valuable in that they can generate any well-behaved function of two inde- pendent variables. They are, how- ever, expensive to build with enough precision, have considerable friction, and take too much space. Bar link- ages are always to be preferred to camoids when it is possible to design such a linkage. 1*8. Integrators. — Integrators are computers that have an output parameter, Xz, and two input parameters, Xi and X2, functionally related by Three-dimensional cam. Xz — Xi = [X2F(X1 JX20 )dX, The simplest form of integrator gives Xz — X{ f KXrfX*. (25) (26) The parameters Xi, X2, of an integrator can be varied at will ; they can, for instance, be given functions of time t The value of the integral, as a function of t, will depend on the form of these functions, and not merely on the instantaneous values of Xi and X2. Thus, unlike a function generator, an integrator does not establish a fixed relation between the instantaneous values of the parameters involved. 24 INTRODUCTION [Sec. 1-8 The equations of integrators are conveniently written in differential form; Eq. (26) becomes then dX, = KX1dX2. (27) This is particularly convenient in schematic diagrams of complete com- puting systems. A common type of integrator is the friction-wheel integrator shown in Fig. 1-25. The output parameter X3 is generated by a friction wheel in contact with a plane disk, the rotation of which is described by the parameter X2. Since the motion of the friction wheel depends on friction between the disk and the wheel, a normal force must act to maintain the Fig. 1-25. — Friction-wheel integrator. frictional force at an adequate level; for this reason the disk is pressed against the wheel by a spring. The friction wheel is transportable along its axis; the distance from the axis of the disk to the point of contact is the parameter X\. In precision integrators the friction wheel is carried by a fixed shaft and the rotating disk is moved with respect to the frame by the amount X\. The equation of the integrator in the figure is dXz = — X\dXi, r (28) where r is the radius of the friction wheel. The double-ball integrator of Fig. 1-26 has the same equation as the friction-wheel integrator ; the difference between these two designs is con- structive only. The friction wheel is replaced by two balls carried in a small cylindrical container, as shown in the figure, or in a special con- Sec. 1-8] INTEGRATORS 25 Fig. 1-26. — Double-ball integrator. Fig. 1*27. — Plan view of component solver. 26 INTRODUCTION [Sec. 1-8 tainer with roller guides for the balls, to reduce friction. These balls transfer the motion of the disk (XidX2) to a drum of radius r, which rotates through an angle dX3 given by Eq. (28). The balls are easily transportable, rolling along the drum, with which they are in contact under constant pressure. This design is useful when one requires an efficient compact computer but does not need the maximum accuracy possible with mechanical integrators. The main source of error is the lack of absolutely sharp definition of the distance from the axis of the plate to the point of contact of the plate with the balls. Any lateral freedom of the lower ball impairs the precision of the results. The component solver shown in Fig. 1-27 is a good example of an inte- grator of the more general type. A large ball of glass or steel is held between four rollers placed in a square, with axes in the same plane, and two rollers with axes parallel to that plane ; the points of contact are at the corners of a regular octahedron. (Figure 1-27 shows only five of the six rollers.) The first four rollers have fixed axes, but the other two have axes that are always parallel, but may assume any direction in the hori- zontal plane. The rotation of these latter axes in the horizontal plane, measured from a certain zero position, is the input parameter X\) the rotation of these rollers on their shafts is the second input parameter X2; the rotation of any one of the four rollers on fixed axes may be taken as an output parameter. Since rollers on parallel axes rotate through equal angles, there are two different output parameters, X3 and X*. If all rollers have the same diameter, the equations of the component solver are dX3 = cos XidX2, (29a) dX4 = sin X,dX2. (296) Thus the component solver is described by Eq. (25), but not by Eq. (26). CHAPTER 2 BAR-LINKAGE COMPUTERS 2*1. Introduction. — A bar linkage is, in the classical sense of the word, a system of rigid bars pivoted to each other and to. a fixed base. In this volume the term "bar linkage" will denote any mechanism consisting of rigid bodies moving in a plane and pivoted to each other, to a fixed base, or to slides. Consideration will be limited to essentially plane mecha- nisms because these are mechanically the easiest to construct. The inclusion in bar linkages of rigid bodies of arbitrary form is not an essential extension of the term, since any rigid body can be replaced by a corresponding system of rigid bars. Similarly, the admission of slides is not a real extension, since bar link- ages— in the classical sense — can be designed to apply the same constraints. A link in a bar linkage is a body « . . .-, i t l Fig. 2-1. — Bar linkage: a nonideal har- COnnected to two Other bodies by monic transformer. pivots. A lever is a body connected to three other bodies by pivots. A crank is a body pivoted to the fixed base, and to one or more other bodies of the linkage. Figure 21 shows a bar linkage that consists of a crank R, a link L, and a slide S. Bar linkages are very satisfactory devices from a mechanical point of view. Pivots and slides are easily constructed and have small backlash, small friction, and good resistance to wear. As computing mechanisms, bar linkages can perform all the functions of the elementary function generators discussed in Chap. 1. They can- not, however, be used to establish relations between differentials; they cannot perform the functions of integrators. As function generators it is characteristic of bar linkages that they do not, generally speaking, per- form their intended operations with mathematical accuracy ; on the other hand, they can generate in a simple and direct way, and with good approx- imation, functions that can be generated only by complicated combina- tions of the classical computing elements. There are few standard bar-linkage function generators; one must usually design a bar linkage for any given purpose. Methods for design- ing such linkages from the mathematical point of view are the main sub- 27 28 BAR-LINKAGE COMPUTERS [Sec. 2-2 ject of this book. The problem is to find a bar linkage that will generate a given function. It must be noted immediately that in general this can be accomplished exactly only by a linkage with an infinite number of elements; mechanisms with a finite number of elements cannot generate the complete field of functions. From a practical point of view, how- ever, even the simpler bar linkages offer enough flexibility to permit solu- tion of the design problem with an acceptably small error. The approach to the problem must be synthetic and approximative, not analytic and exact. The mechanical design of bar linkages cannot be discussed in this volume. It is of course possible to treat analytically the properties of a given linkage : its motion, the distribution of velocities of its parts, acceler- ations, inertia, forces. In this respect the theory of linkages has been well developed, even in elementary texts; the kinematics of bar linkages have been treated especially thoroughly. It is of course necessary that the designer of linkages have knowledge of the practical properties of these devices, even when he is primarily interested in their mathematical design. In the present volume there will be some comment on the mechanical features of bar linkages, but only enough to give the designer the necessary base for reasoning when the design procedure is started. 2*2. Historical Notes. — Engineers and mathematicians have in the past considered bar linkages primarily as curve tracers — that is, as devices serving to constrain a point of the linkage to move along a given curve. The classical problem in the field has been that of finding a bar linkage that will constrain a point to move along a straight line. This problem was consid- ered by Watt in designing his steam engine. Watt found a sufficiently accurate solution of the problem, and it was the cost and space required that caused the use of a slide in his original design. Bar linkages are now extensively used in mechanical design because of their small frictional losses and high efficiency in transmitting power — efficiency greater than that of any gear or cam. The usefulness of bar linkages to the mechanical engineer can be illustrated by a locomotive: its transmission contains the famous parallelogram linkage, and the valve motions are controlled by bar linkages of some complexity. A designer of linkage multipliers will :! u m m- Fig. 2-2. T -Bar linkages in a microscope plate holder. Sec. 2-2] HISTORICAL NOTES 29 recognize among these structures elements that he is accustomed to use in his own work. Bar linkages are used in heavy construction as counterweight linkages and for the transmission of spring action. They also serve as elements of fine instruments. The parallelogram linkage used to assure pure trans- lational motion of a slide being examined by a microscope is illustrated in Fig. 2-2. Springs are omitted from the diagram. The field of the micro- scope is indicated at the center of the plate. The problem of producing an exact straight-line motion by a bar linkage was first solved by Peaucellier. x This was accomplished by application of the Peaucellier inversor to the conversion of the circular motion of a crank into a rectilinear motion. The Peaucellier inversor is illustrated in Fig. 2-3. It consists of a jointed quadrilateral with four sides of equal lengths B, to the oppo- site vertices of which there are jointed two other bars of equal lengths A; these latter bars are themselves joined at their other ends. Three joints of this structure neces- sarily lie on the same straight line, and the distances Xi and X2 between these joints vary inversely with each other. It will be noted that Xi is the sum of the lengths of the bases of two right triangles of altitude T and hypotenuses A and B respectively, whereas X2 is the difference of these base lengths. We have then .-' S Fig. 2-3. — Six-bar Peaucellier inversor. The solid lines illustrate the case B < A, the dashed lines the case B > A. Z2 = VA2 - P - \/B* - T\ (la) (16) In these equations A, B, T, and the square roots are necessarily positive. On multiplying together Eqs. (la) and (16) we obtain XXX2 = A2 - B2, (2a) or A2 - Bi X, (26) There are two variants of this inversor, with A greater than B or with B greater than A. If B is greater than A (dashed lines in Fig. 2-3), X2 is always negative; there is no possibility of having Xi equal X2. If A is greater than B (solid lines in Fig. 2-3), it is possible to have X1 = X2 = (A2 - B2)* 1 A concise summary of work in this field, by R. L. Hippisley, will be found under Linkages, in the Encyclopedia Britannica, 14th ed. 30 BAR-LINKAGE COMPUTERS [Sec. 2-2 At this point the mechanism exhibits an undesirable singularity; the joints P and Q of Fig. 2-3 become coincident, and self-locking of the device may occur. These two forms of the Peaucellier inversor also differ in their useful ranges. These are VA2 - B2 < Xi < A + B, tiA>B, (3a) B-A<Xl<A+B, UA<B. (36) The freedom from self-locking and the greater range make it desirable to have B greater than A. Figure 24 shows the Peaucellier inversor in a form suitable for use as a computer. U V M Xi ^B ( ^-tf/( v*r *T-"~^ Fig. 2-4. — Three-bar Peaucellier inversor. Fig. 2-5. — The Hart inversor. Another inversor has been devised by Hart.1 The Hart inversor (Fig. 2-5) is essentially a bar-linkage parallelogram with one pair of bars reflected in a line through opposite vertices. Let any line OS be drawn parallel to a line UV through alternate vertices of the quadrilateral. It can be shown that this will intersect adjacent bars of the linkage at points 0, P, Q, that remain collinear as the linkage is deformed; furthermore, the distances Xi = OQ and X2 = OP will vary inversely with each other. There have been described linkages for the tracing of conic sections, the Cassinian oval, the lemniscate, the limacon of Pascal, the cardioid, and the trisectrix; indeed it is theoretically possible to describe any plane curve of the nth degree in Cartesian coordinates x and y by a bar linkage.2 Linkages for the solution of algebraic equations have also been devised.3 1 H. Hart, "On Certain Conversions of Motion," Messenger of Mathematics, 4, 82 (1875). 2 A. Cayley, "On the Mechanical Description of a Cubic Curve," Proc. Math. Soc, Lond., 4, 175 (1872). G. H. Dawson, "The Mechanical Description of Equipotential Lines," Proc. Math. Soc, Lond., 6, 115 (1874). H. Hart, "On Certain Conversions of Motion," Messenger of Mathematics, 4, 82 and 116 (1875); "On the Mechanical Description of the Limacon and the Parallel Motion Deduced Therefrom," Messenger of Mathematics, 5, 35 (1876); "On Some Cases of Parallel Motion," Proc. Math. Soc, Lond., 8, 286 (1876-1877). A. B. Kempe, "On a General Method of Describing Plane Curves of the nth Degree by Linkwork," Proc. Math. Soc, Lond., 7, 213 (1875); "On Some New Linkages," Messenger of Mathematics, 4, 121 (1875). W. H. Laverty, "Extension of Peaucellier's Theorem," Proc. Math. Soc, Lond., 6, 84 (1874). 3 A. G. Greenhill, "Mechanical Solution of a Cubic by a Quadrilateral Linkage," Messenger of Mathematics, 5, 162 (1876). A. B. Kempe, "On the Solution of Equa- tions by Mechanical Means," Messenger of Mathematics, 2, 51 (1873). Sec. 2-3] DESIGN OF BAR-LINKAGE COMPUTERS 31 Analytical studies1 have been made of the " three-bar motion" of a point C rigidly attached to the central link AB of a three-bar linkage (Fig. 2-6). Three-bar motion is very useful in the design of complex computers, and will be discussed in Sec. 10-4. To complete this survey of the bar-linkage literature in English, it will suffice to mention the papers of Emch and Hippisley on closed linkages.2 2*3. The Problem of Bar -linkage-computer Design. — It is only recently that much attention . has been paid to the problem of using bar linkages . FlG- 2-6— Three-bar in computing mechanisms. The literature in the rigidly attached to the field is especially restricted. The author knows of central bar. only one published work that employs the synthetic approach to bar- linkage computer design3 — and this in a more restricted field than that of the present volume. The basic ideas in the synthetic approach to bar-linkage design are simple, but quite different from the ideas behind the classical types of computers. Bar linkages can be characterized by a large number of dimensional constants, and the field of functions that they can generate is correspondingly large — though not indefinitely so. Given a well- behaved function of one independent variable, one should be able to select from the field of functions generated by bar linkages with one degree of freedom at least one function that differs from the given function by a relatively small amount. The characteristic problem of bar-linkage design is thus that of selecting from a family of curves too numerous and varied for effective cataloguing one that agrees with a given function within specified tolerances. The presence of a residual error sets bar linkages apart from other computing mechanisms. The error of a computer of classical type arises from its construction as an actual physical mechanism, with unavoidable imperfections. It is possible to reduce the error to within almost any limits by sufficiently careful design — as, for instance, by enlarging the 1 A. Cayley, "On Three-bar Motion," Proc. Math. Soc, Lond., 7, 136 (1875). R. L. Hippisley, "A New Method of Describing a Three-bar Curve," Proc. Math. Soc, Lond., 15, 136 (1918). W. W. Johnson, "On Three-bar Motion," Messenger of Mathe- matics, 5, 50 (1876). S. Roberts, "On Three-bar Motion in Plane Space," Proc. Math. Soc, Lond., 7, 14 (1875). 2 A. Emch, "Illustration of the Elliptic Integral of the First Kind by a Certain Link-work," Annals of Mathematics, Series 2, 1, 81 (1899-1900). R. L. Hippisley, "Closed Linkages," Proc. Math. Soc, Lond., 11, 29 (1912-1913); "Closed Linkages and Poristic Polygons," Proc Math. Soc, Lond., 13, 199 (1914-1915). 3 Z. Sh. Blokh and E. B. Karpin, "Practical Methods of Designing Flat Four-sided Mechanisms," Izdatelstvo Akademie nauk SSSR, Moscow, Leningrad (1943). E. B. Karpin, "Atlas of Nomograms," Izdatelstvo Akademie nauk SSSR, Moscow, Lenin- grad (1943). 32 BAR-LINKAGE COMPUTERS [Sec. 2-4 whole computer. In bar linkages there is usually a residual error that cannot be eliminated by any care in construction, an error that is evident in the mathematical design of the device, as well as in the finished product. This error will be called " structural error" because it depends only on the structure of the computer, and not on its size or other mechanical proper- ties. Reduction of structural error requires a change in the structure of the computer — usually the addition of parts. The great number of adjustable dimensional constants gives greater flexibility and extends the field of functions that the linkage can generate; from this larger field of functions one can then select a better approximation to the given function. The fact that bar linkages can be used to generate functions of a large class has been known for many years, and has been used (instinctively, rather than with a full development of the theory) by designers of mecha- nisms. The field of functions that can be generated by some simple bar linkages has been analytically described. This, however, represents only the easier half of the problem; what one needs is to describe the field of functions that can almost be generated by a given type of linkage. The first attempts to solve this problem for one independent variable have been tabular or graphical. For very simple structures it is possible to devise graphs that allow one to determine whether a given function can be generated approximately by such a structure, and what structural error is inevitable. These methods are practicable if the linkage can be specified by means of only two dimensional parameters — that is, if the field of functions depends upon only two adjustable parameters. Such graphical methods are difficult or are necessarily incomplete if the field of functions depends upon three adjustable parameters. Such a procedure can hardly be attempted when four or more dimensional parameters are involved. The design methods presented in this book are in many cases based on a graphical factorization of the given function into functions suitable for mechanization by simple linkages; the elements of the mechanism designed in this way can then be assembled into the desired complete linkage. By such methods it is possible to design linkages having a great many adjustable parameters, but the solution obtained cannot be claimed to be the best possible. Usually it is easy to apply these methods to find bar linkages that have errors everywhere within reasonable tolerances. This is ordinarily sufficient for practical purposes. 2-4. Characteristics of Bar -linkage Computers. — The special proper- ties of bar-linkage computers may be summarized as follows. Advantages. 1. Bar linkages occupy less space than classical types of computers. 2. They have negligible friction. Sec. 2-4] CHARACTERISTICS OF BAR-LINKAGE COMPUTERS 33 3. They have small inertia. 4. They have great stability in performance. 5. Their complexity does not necessarily increase with the complexity of the analytical formulation of the problem. 6. They are easy to combine into complex systems. 7. They are relatively cheap. Disadvantages. 1. Bar linkages usually possess a structural error. 2. The field of mechanizable functions is somewhat restricted. 3. The complexity of the linkage increases with decreasing tolerances. 4. Linkage computers are relatively difficult to design. The difficulty of the design procedure increases with increasing complexity and decreasing tolerances. 5. The travel of the mechanism is usually limited to a few inches. Backlash error and elasticity error must be reduced by careful construction : the use of ball bearings is essential, and rigidity of the structure perpendicular to the plane of motion must be assured. The design should be such that mechanical errors are less than the assigned tolerances for structural error. Bar linkages can attain extensive use as elements of computers only as efficient methods of design are established. The complexity and difficulty of the design procedure depends largely on the nature of the given func- tion. It is usually easy to design a linkage with a structural error that does not exceed 0.3 per cent of the whole range of motion of the computer. It becomes relatively laborious to reduce the structural error below 0.1 per cent. If the tolerances are below 0.1 per cent — as a typical value — alternatives to the use of a bar linkage should be explored. Bar linkages can advantageously be combined with cams when the tolerated error is small and a bar linkage alone would be excessively com- plex. For instance, if a given function of one independent variable were to be mechanized with an error of not more than 0.01 per cent, it might be desirable to mechanize this function by a simple bar linkage with an error of, for example, 1 per cent, and to use a cam to introduce the required correction term. Since this corrective term represents only 1 per cent of the whole motion of the linkage, it need not be generated with very high precision; for instance, if the working displacement of the cam is to be 1 in., it can be fabricated with a tolerance as rough as 0.01 in. It is a feature of bar-linkage computers that they can be used to generate functions of two independent variables in a very direct and mechanically simple way. Methods for the design of linkages generating functions of three independent variables are not now available when it is 34 BAR-LINKAGE COMPUTERS [Sec. 2-5 not possible to reduce the problem to the mechanization of functions of one or two independent variables; there is, however, some hope that practically useful methods can be found. Bar-linkage computers have great advantages when feedback is to be used in the design of complex computers. In computers of the classical type, feedback motion must be a small fraction of the total output motion. Linkage computers can, however, operate very close to the critical feed- back— that is, the degree of feedback at which the position of the mecha- nism becomes indeterminate. 2-5. Bar Linkages with One Degree of Freedom. — Bar linkages with one degree of freedom serve the same purpose as cams; they may be called " linkage cams." The parallelogram linkage of Fig. 2-2 and the linkage inversors have motions expressed accurately by very simple formulas, but they are not generally useful in the mechanization of given functions. For this purpose, the following bar linkages are much more interesting. The harmonic transformer, shown in Fig. 2-1, establishes a relation between an angular parameter Xi and a translational parameter X2. It is convenient to disregard variations in the form of this relation due to changes in scale of the mechanism — to consider as equivalent two geo- metrically similar mechanisms. The field of functions X2 = F(X0 (4) generated by the harmonic transformer then depends upon two ratios of dimensions: L/R and E/R, the ratios to the crank length of the link length and the displacement of the crank pivot from the center line of the slide. As L is increased from its minimum value, the plot of X2 against Xi changes (in a typical case) from an isolated point to a closed curve, then to a sinusoid, and finally, in the limit as L approaches infinity, to a pure sinusoid. From a practical point of view, the pure sinusoidal form is reached for links short enough for practical use. In the limiting case, L = » , the equation of the harmonic transformer is X2 = R sin X1 + C. (5) Such a harmonic transformer will be called "ideal." Only rarely is the complete range of motion of a harmonic transformer used. When the range of the parameter X\ is limited to X\m < X\ < Xim and the functions defined within these restricted limits are taken as ele- ments of a new functional field, there is obtained a four-dimensional functional field depending on X\m and XXM as well as on L/R and E/R. Methods for the design of harmonic transformers will be discussed in Chap. 4. Sec. 2-5] BAR LINKAGES WITH ONE DEGREE OF FREEDOM 35 The three-bar linkage shown in Fig. 2-7 consists of two cranks pivoted to a frame and joined at their free ends by a connecting link. As a computer, this serves to "compute" the parameter X2 as a function of the parameter X\. The linkage itself is described by four lengths : A\f Bit A2, B2. The field of functions generated by this type of linkage is only three-dimensional, because two geometrically similar mechanisms estab- Fig. 2-7. — Three-bar linkage. Pig. 2-8. — Three-bar linkage modified by eccentric linkage. lish the same relation between Xi and X%. The field of functions thus depends on three ratios — for example, Bi/Ai, A2/A1, and B%/A\. Usu- ally only a part of the possible motion of the mechanism is used. Limits of motion can be assigned for Xi or X2, though, of course, not independ- ently for the two parameters; for instance, one may fix X\m < X\ < X\m. This increases the number of independent parameters by two ; the field of functions generated by a three-bar linkage operating within fixed limits r-\^-^-^±r Fig. 2-9. — Harmonic transformer modified by eccentric linkage. is five-dimensional. In Chap. 5 we shall see how to design a three-bar linkage for the approximate generation of a given function. The eccentric linkage is not a bar linkage, but is so conveniently used in connection with bar linkages that it should be mentioned here. Figure 2-8 shows a three-bar linkage modified by the insertion of an eccentric linkage. One crank of the three-bar linkage carries a planetary gear that meshes with a gear fixed to the frame. The central link is then pivoted eccentrically to the planetary gear, rather than to the crank itself. Link- ages of this type will be discussed in Sec. 7-9, where their importance will 36 BAR-LINKAGE COMPUTERS 40 30 >ec. 2-5 Fig. 2-10. — Double three-bar linkage generating the logarithmic function- be explained. Another important application of the eccentric linkage is in the modification of harmonic transformers, as illustrated in Fig. 2-9. It is possible to choose the constants of the eccentric linkage in such a way that the linkage output is an almost perfect sinusoid, even though the length of the link L is relatively small. Combinations of these linkages to be discussed in this book are the double harmonic transformer (Sec. 4-9 and following), harmonic transformers in series with three- bar linkages (Sec. 8-1 and following), and the double three-bar linkage (Sec. 8-8). Figure 2-10 shows a double three-bar linkage that generates the logarithmic function through the range indicated in the figure. Fig. 2-11. — Bar linkage with two degrees of freedom. Sec. 2-6] BAR LINKAGES WITH TWO DEGREES OF FREEDOM 37 2'6. Bar Linkages with Two Degrees of Freedom. — Bar linkages with two degrees of freedom can be used in the generation of almost any- well-behaved function X3 = F(Xlf X2) (6) of two independent variables. They provide a mechanically satisfactory substitute for three-dimensional cams, which have many disadvantages and are to be avoided if possible. Figure 2-11 shows a linkage with two degrees of freedom, which consists of three cranks connected by two links and a lever. The lever will degenerate into a simple link if the pivots A and B are superposed; the resulting struc- ture of three links jointed at a single pivot will be called a "star linkage." Its properties are dis- cussed in Chap. 9. The bar-linkage adder shown in Fig. 2-12 consists of essentially the same parts as the linkage of Fig. 2-11, except that slides are used instead of cranks to constrain the links. The dimensions obey the simple relation Fig. 2-12. — Bar-linkage adder. At A2 Bi = Ci B2 C2 (7) It is easy to show that when this proportionality holds, the three pivots Pi, P2, and P3 lie on a straight line. This device can, therefore, be used to mechanize any alignment nomogram that consists of three parallel straight lines; in particular, it can be used to mechanize the well-known nomogram for addition. If Xh X2, and X3 are three parameters measured along these lines in the same direction from a common zero line, then (Ai + A2)Xs = AiZi + A2X2. (8) This bar linkage is free from structural error. In contrast to the adders, bar-linkage multipliers do not perform the operation of multiplication exactly, but with a small error; the equation of such a multiplier is RXZ = XXX2 + 5, (9) where 5, the error of the multiplier, is a function of the two independent parameters Xi and X2. The design of multipliers will be discussed in Chap. 9; a much simplified explanation of the principle will be given here. 38 BAR-LINKAGE COMPUTERS [Sec. 2-6 Figure 2-13 shows the essential elements of one type of multiplier. Three bars of equal lengths, Ri = R2 = Rz = I, are pivoted together. The first is pivoted also to the frame at the point 0, the third to a slide with center line passing through 0. If the joints A\ and A2 are placed at distances Xi and X2 from the center line of the slide, the distance OS = D will be exactly d = vi - xi - Vi - (** - *i)2 + Vi - x\ Expanding in series the terms on the right, one obtains X, = 1 - D = XXX2 + iXiXf - fX?Xl + -IXIX2 + • (10) , (11) where X; Fig. 2-13. -Elements of multiplier. bar-linkage is the displacement of the pivot S from the position S0 which it occupies when Xi = X2 = 0 and the three links are coincident. It is evident that X3 is equal to the product XiX2 to the approxima- tion in which the terms of fourth and higher degrees can be neg- lected in comparison with the term of the second degree. For sufficiently small values of Xi and X2 this mechanism is thus a multiplier for these parameters. Such a multiplier is not practical, however, because of its small range of motion. If the error in the multiplication is to be kept below 1 per cent, it is necessary to keep Xi, X2 ^ 0.2. [If Xi = X2 = 0.2, then X3 = (0.2)2 + i(0.2)4 + * ' * , and the fractional error is almost exactly one per cent.] Under these conditions, however, one has X3 = 0.04, an impracticably small range of motion. There are in principle two ways to improve this multiplier. With either method it is necessary to make the structure more complicated — to add new adjustable parameters. One possible arrangement is indicated in Fig. 2-14. Here the parameter X2 is a displacement of a slide (of adjustable position) that controls the position of the joint ^42 through a link of adjustable length L2; X3 becomes an angular parameter, the angle turned by a crank with adjustable length and pivot position. With the first method, the output parameter X3 is expressed in terms of Xi and X2, in the form of a series with coefficients which depend on the adjustable dimensions of the mechanism. These dimensions can then be so chosen as to cause the terms of the fourth degree in Xi and X2 Sec. 2-6] BAR LINKAGES WITH TWO DEGREES OF FREEDOM 39 to vanish. In this way, the multiplier can be made more accurate for small values of Xi and X2, and the domain of useful accuracy sub- stantially increased. Toward the limits of this domain, however, the inaccuracy of the multiplier will increase very rapidly. The second method for im- proving the multiplier — that followed in this book — can be indi- cated only very roughly at this point. It involves comparison of the ideal product and the function actually generated by the multi- plier over the entire range of motion, and adjustment of the dimensional constants of the sys- tem in such a way that the error of the mechanism is brought within specified tolerances everywhere within this domain. To see in principle how this can be done, let us consider the mechanism of Fig. 2-13. Let Xz and Xi be given a series of values that have the fixed ratio Fig. Modified bar-linkage multiplier. = xj. (12) If this linkage were an exact multiplier, the pivot A2 would indicate always the same value of X2; it would move along a straight line at constant distance X2 from the line of the slide. Actually, the pivot A 2 will describe a curve that is tangent to this straight line for small values of Xi and X3, but will diverge from it as these parameters increase. To each value of X2 there will correspond another curve ; the curves of constant X2 form a family, each of which can be labeled with the associated value of this parameter. Now we can make this multiplier exact if we can introduce a constraint which, for any specified value of X2, will hold the pivot A2 on the corresponding curve of this family. For example, if these curves were all circles with the same radius L2 and centers lying on a straight line, it would be possible to use the type of constraint illustrated in Fig. 2-14. The X2-slide could then be used to bring the pivot A3 to the center of the circle corresponding to an assigned value of X2, and the pivot A 2 would stay on that circle, as required. Actually, the curves of con- stant X2 will not form such a family of identical circles. It will, however, be possible to approximate them by such circles in a way which will split the error and bring it within tolerances held fairly uniformly over the whole domain of action. Unlike the multipliers designed by the first method, a multiplier thus designed will not have unnecessarily small errors in one part of the domain and excessively large errors in another part. 40 BAR-LINKAGE COMPUTERS [Sec. 27 This concept of multiplier design must be very greatly extended before it can lead to the design of satisfactory computers. A powerful guide in beginning the work is provided by the idea of nomographic multipliers, already discussed in Sec. 1*5. It is possible to design approximate inter- section nomograms for multiplication that have as their mechanical analogues bar linkages with two degrees of freedom. For instance, Fig. 8-14 shows a nomogram for multiplication obtained by topological trans- formation of the nomogram of Fig. 1-12; it consists of two families of identical circles and a third family of curves that can be very closely approximated by a family of identical circles. This nomogram cor- responds to the bar-linkage multiplier illustrated in Fig. 8-15, which, on improvement of its mechanical features, takes on the form shown in Fig. 8-16. The design techniques to be described in Chaps. 8 and 9 make it possible to design multipliers with large domain of action and good uniformity of performance through this domain. Multipliers can be used to perform the inverse operation of division; that is, they can be used to evaluate X2 = X3/X1. It is, of course, not possible to divide by zero; when a multiplier is used in this way Xi will never pass through zero. It is therefore useless to attempt to reduce to zero the error of such a multiplier for values of Xi very near to zero; it is also undesirable to attempt to reduce the errors of the device for nega- tive values of Xi when only positive values can be introduced. For this reason three types of multiplier may be distinguished. 1. Full-range multipliers, for which both input parameters can change signs. 2. Half-range multipliers, for which only one parameter can change signs. 3. Quarter-range multipliers, for which neither input parameter can change signs. Dividers may be divided into two types. 1. The plus-minus type, for which the numerator may change sign. 2. The single-sign type, for which all parameters have fixed signs. An example of a practical full-range linkage multiplier is shown in Fig. 8-16; a half -range multiplier is shown in Fig. 9-15. 2-7. Complex Bar -linkage Computers. — The elementary linkage cells already described may be combined to form complex computers. Since simple linkages can add, multiply, and generate functions of one and two independent variables, bar-linkage computers can solve any problem that can be expressed in a system of equations involving only these operations. The field of application of bar-linkage computers is quite large; they Sec. 2-7] COMPLEX BAR-LINKAGE COMPUTERS 41 are especially useful if the computer must be light, as when it is to be carried in aircraft or guided missiles. An important feature of bar-linkage computers is the ease with which the cells can be assembled into a compact unit. It is natural to spread the parts of the computer out in a plane, to produce a rather flat mecha- nism with its parts easily accessible. The connections between cells are provided by shafts or connecting bars. There is a simple trick that makes the connection of linkage cells even easier, and the structure of some cells less complex. The simplification of linkage adders is a characteristic example of this trick. The bar-link- age adder shown in Fig. 2*12 has no structural error. Any deviation from the principle of this design is likely to lead to a structural error; it is, how- ever, possible to change the principle in such a way that the structural error x2 is negligibly small. For instance, if Fl°- 21 5.— Bar-linkage adder (approxi- the links Bi and B2 are very long, their lengths can be chosen at will without appreciably affecting the accuracy of the addition. Figure 2-15 shows such an approximate adder; its equation is (Ai + A2)X3 « A 1X1 + A2X2. (13) The links Li and L2 must be so long that they lie nearly parallel to the lines of the slide, but they need not be exactly parallel to each other. The action of this device depends upon the essential constancy of the projec- tion of the lengths of these bars along the line of the slides. Let X\, X2, and X'z be defined as the distances of the pivots Pi, P2, and P3 from some zero line perpendicular to the line of the slides. One then has, exactly, (Ax + A2)Xi - AxX; + A2X2. (14) Now let 0i be the angle between the bar L\ and the line of the slides. Then Xi = X[ + Lx cos 0i + C, (15a) = Xi - Li(l - cos 0i) + (C + Li). (156) Except for an additive constant (which can be reduced to zero by proper choice of the zero point), X[ and Xt differ only by the variable term Li(l — cos 0i). As Li is increased, 0i decreases with 1/Xi, (1 — cos 0i) decreases with 1/Lf, and Li(l — cos 0i) decreases with 1/Li. Thus, by making L\ large and properly choosing the zero point, one can make Xi and X[ differ by a negligibly small term. In the same way X2 can be made negligibly different from X2; X3 and X3 are identical. Equation 42 BAR-LINKAGE COMPUTERS [Sec. 2-7 (13) follows as an approximation to Eq. (14). If 0i is kept less that 0.035 radians (about 2°) the difference between X\ and X[ will be about 0.0006 L\. Thus if the bars deviate from parallelism with the slides by no more than + 2° during operation of the adder, the resulting error in the output will not exceed 0.06 per cent of the total length of the bars. If the lengths of the bars in approximate adders are great enough, it is even immaterial whether the slides move along straight lines ; the essential 7777777 Fig. 2-16. — Combination of approximate adders. thing is that the parameters be measured as distances from a zero line. It is, therefore, possible to connect adding cells through long connecting bars, and to omit some of the slides that would appear in the standard construction. Fig. 2-16 shows a combination of three adding cells that will solve (approximate!}7) the equations (A1 + A2)XS = A1X1 + A2X2, (D4 + Db)X1 = D4X4 + D,X5, (E$ + Eg)X7 = E5X5 + E&Xe. (16) CHAPTER 3 BASIC CONCEPTS AND TERMINOLOGY The present chapter will define the terminology to be employed in discussing bar-linkage design and introduce some concepts with wide application in the field. Of particular importance are the concepts of " homogeneous parameters" and " homogeneous variables/' and a graphi- cal calculus used in discussing the action of computing mechanisms in series. 3*1. Definitions. Ideal Functional Mechanism. — Any mechanism can be used as a computer if it establishes definite geometrical relations between its parts — that is, if it is sufficiently rigid and free from backlash, Zero position 1 — h 1- ' '1 r^i !___ Ql xj ,1 j^j -J L Fig. 3-1. — Crank terminal. Fig. 3-2. — Slide terminal. slippage, or mechanical play. In the following discussion we shall be concerned only with such ideal functional mechanisms. Terminals. — The terminals of a computing mechanism are those ele- ments that, by their motions, represent the variables involved in the computation. The motion of all terminals is usually specified with respect to some common frame of reference. If the position of a terminal is controlled in order to fix the configuration of the mechanism, it may be called an " input terminal"; if its position is used in controlling a second mechanism, or is simply observed, it may be called an " output terminal." A terminal may be suitable for use only as an input terminal, or only as an output terminal, or in either way, according to the nature of the mechanism. 43 44 BASIC CONCEPTS AND TERMINOLOGY [Sec. 31 Terminals that are mechanically practical are of two kinds: 1. Crank or rotating-shaft terminals (Fig. 3-1), which represent a variable by their angular motion. 2. Slide terminals (Fig. 3-2), which represent a variable by a linear motion. Parameters. — A parameter is a geometrical quantity that specifies the position of a terminal. With a crank terminal, it is usually the angular position of the terminal with respect to some specified zero position; with a slide terminal, it is usually the distance of the slide from a zero position. Parameters may be defined in other ways — for instance, as the distance of a slide terminal from some movable element of the mechanism — but such parameters are less generally useful than those just mentioned. An input parameter describes the position of an input terminal, an output parameter that of an output terminal. Linkage Computers. — A linkage computer establishes between its parameters, X\, X2, . . . X8, definite relations of the form Fr(X1( X„ • • • X.) = 0, r = 1, 2, • • • , (1) which involve only these parameters and the dimensional constants of the mechanism. With more general types of mechanisms these equations of motion may also involve derivatives of the parameters. Such mecha- nisms are useful in the solution of differential equations, but they will be excluded from our future considerations; we shall be concerned only with linkage computers, which generate fixed functional relations between the parameters. To describe the configuration of linkage computers with n degrees of freedom, one must in general specify the values of n input parameters, Xi, X2, . . . Xn. The values of any number of output parameters can then be expressed explicitly in terms of these n parameters: Xn+r = Gr (Xh X2. . ■ • Xn), r = 1, 2, ■ • • m. (2) Domain. — The parameters of a computing mechanism cannot, in general, assume all values. The limitations may arise from the geometri- cal nature of the mechanism (a linear dimension will never change without limit) or from the way in which it is employed. To each possible set of values of the input parameters Xh . . . Xn, there corresponds a point (Xi, X2, . . . Xn) in n-dimensional space; to all sets of values that may arise during a specific application of the mechanism, there corresponds a domain in n-dimensional space, which will be referred to as the "domain" of the parameters. It must be emphasized that the domain of the param- eters is not necessarily determined by the structure of the mechanism, but by the task set for it. Sec. 31] DEFINITIONS 45 In the most general case, the domain of the input parameters may be of arbitrary form — except, of course, that it must be simply connected, since all parameters must change continuously. In such cases the values possible for any one parameter may depend on the values assigned to other parameters. A mechanism will be said to be a " regular mechanism' * when each input parameter can vary independently of all others, between definite upper and lower limits, Xim ^ Xi ^ Xui, i - 1, 2, • • • n, (3) which define the domain of the parameter. With angular parameters, neither of these limits is necessarily finite : it is possible to have Xim = — «' , or XiM = + °° . The output parameters of a regular mechanism will vary between definite (though not necessarily finite) limits as the input parameters take on all possible values. These limits serve to define a domain for each output parameter. Although the input parameters vary inde- pendently through their respective domains, this is not always true of the output parameters. Travel. — The range of motion of a terminal is called its "travel." This is AX, = XiM - Xim, (4) both for input and output terminals. Variables. — The term "variable" will denote the variables of the problem which the computing mechanism is designed to solve. A varia- ble will be associated with each terminal of a mechanism, an input variable with an input terminal, an output variable with an output terminal. To each value of a variable there will correspond a definite configuration of the terminal ; each variable, then, will be functionally related to a param- eter of the mechanism : Xi = UXi). i = 1, 2, • • • . (5) It is important to keep in mind the distinction between parameters, which are geometrical quantities measured in standard units, and the variables of the problem, which are only functionally related to the param- eters. In this book, variables will be denoted by lower-case letters, parameters by capitals. Scales. — The value of the variable corresponding to a given configura- tion of a terminal can be read from a scale associated with that terminal. The calibration of this scale is determined by the form of the functional relation between Xi and Xi. If as, is a linear function of Xi the scale will be even — that is, evenly spaced calibrations will correspond to evenly spaced values of as,-. Such a scale may also be referred to as "linear," 46 BASIC CONCEPTS AND TERMINOLOGY [Sec. 31 in reference to the form of the functional relation represented. (This term does not describe the geometrical form of the scale, which may be circular.) A linear terminal is a terminal with which there is associated a linear scale. Range of a Variable. — As a parameter changes between its limits, Xim and XiM, the associated variable will also change within fixed, but not necessarily finite, limits: Xim ^ Xi ^ XiM. (6) In the case of a regular mechanism, this may be referred to as the " domain" of the variable; its range is Azt = xiM — xim. (7) Mechanization of a Function. — An ideal functional mechanism estab- lishes definite relations between its parameters: Fr(Xh X2, • • • ) = 0, r = 1, 2, • • • . (8) It may be said to provide "a mechanization" of these functional relations within the given domain of the independent parameters. Such a mechanism, together with its associated scales, similarly pro- vides a mechanization of functional relations, Mxh xit - - • ) = 0, r = 1, 2, • • • , (9) between the variables xi} within a given domain of the independent varia- bles. The forms of these relations may be derived by eliminating the values of the parameters Xi between Eq. (8), which characterizes the mechanism, and Eq. (5), which characterizes the scales. 1 2 3 45 T-i H tl-tH 9 fl 8 7 6 ft j-@ — " ©-. Ss <°r Fig. 3-3. — Input scale. If the output variables are to be single-valued functions of the input variables, the input parameters must be single-valued functions of the input variables, and the output variables must be single- valued functions of the output parameters; it is not, however, necessary that the inverse relations be single- valued. Thus an input scale may have the form shown in Fig. 3-3, and an output scale that shown in Fig. 3*4, but not the reverse. Linear Mechanization. — A mechanization of a relation between varia- bles will be termed a "linear mechanization" if all scales are linear. Sec. 3-2] HOMOGENEOUS PARAMETERS AND VARIABLES 47 A nonlinear mechanization of a given function may be useful when input variables are set by hand, and only a reading of the output variables is required. When a computing mechanism is to be part of a more com- plex device, it is usually necessary that the terminals have mechanical motion proportional to the change in the associated variable — that is, a linear mechanization of the function is needed. For instance, if one has only to com- pute the superelevation angle for an antiaircraft gun it may be quite satis- factory to read this on an unevenly divided scale. If, however, one wishes to use the computer to control directly the sight on a gun, then a linear mecha- nization of the superelevation function will be required. It is a trivial matter to design a nonlinear mechanization of a function of one independent variable. One requires only a single pointer, serving both as input and output terminal, to indicate corresponding values of input and output variables as parallel scales (Fig. 3-5). For this reason the term mechanization as applied to functions of a single independent variable will always denote linear mechanization; a distinction will be made between linear and nonlinear mechanization only in the case of linkages of two or more degrees of freedom. Fig. 3-4. — Output scale. i '■ i ■ ■ ■ t ■ ■^r 3 3 4 4 5 "l'"T"T"T" 6 7 8 9 Fig. 3-5. — Nonlinear mechanization of a function of one independent variable. 3-2. Homogeneous Parameters and Variables. — Homogeneous vari- ables and parameters are very useful tools in the design of individual computing linkages, and also in the drawing up of schematic diagrams for complex computers. They are defined only for variables and parameters which vary within finite limits. Associated with each variable xt having a finite range Axi is a homo- geneous variable defined by h = Xj ■ XiM Xim X !n (10) 48 BASIC CONCEPTS AND TERMINOLOGY [Sec. 3-2 As Xi varies from its lower to its upper bound, h varies linearly with it, from 0 to 1. The inverse form of Eq. (10) may be written Xi = Xim + hiAXi. (11) Another homogeneous variable, "complementary to hi," is denned by hi - Xi*~Xi, (12) X\M Xim or by hi + hi = 1- (13) In the same way, there are associated with each parameter Xi} having a finite travel AX,-, two complementary homogeneous parameters, tt •**■* **-im / 1 a \ Hi = ■= y~, (14) •A-iM -Aim Hi = l- Hi} (15) which change linearly with X, between bounds 0 and 1 : Xi = Xim + HiAXi = Xi,t - HiAXi. (16) In a linear mechanization, the homogeneous variables and parameters are very simply related. The quantities X{ and xx are connected by a linear relation, Xi - XT = Uxi - af>). (17) If ki is positive, the minimum values of X, and x> occur together, as do the maximum values: Xim - Xf> = ki(xim - aj»), (18a) (h > 1) XiM - Xf = ki(xiM - xf). (186) It follows by introduction of these relations into Eqs. (10) and (14) that Hi m h. (h > 1). (19) If ki is negative, the maximum value of Xi occurs together with the mini- mum value of Xi, and conversely: Xim - X^ = ki(xtM - xf), (20a) (h < 1) XiM- XT = ki(xim-xT); (20b) then Hi m fit - 1 - hi. (h < 1). (21) Equation (19) will be referred to as the "direct" identification of Hi with hi. It implies that X, and Xi are linearly dependent on each other, Sec. 3-3] AN OPERATOR FORMALISM 49 changing in the same sense between minimum and maximum values which they attain simultaneously ; the scale of Xi is even, and increases in the direction of increasing Xt. Equation (21) will be termed the "com- plementary identification" of Hi and hi) it implies that the scale of Xi is even, and increases in the direction of decreasing Xi. In terms of homogeneous variables, the problem of linearly mechaniz- ing a given function takes on a particularly simple form. For instance, if the given function involves a single independent variable, it may be expressed in terms of a homogeneous input variable hi and a homogeneous output variable h2: h2 = f(hi). (22) A linkage with one degree of freedom, operating in a specified domain of the input parameter, Xim ^ Xi ^ XiMi (23) will generate a relation between homogeneous input and output param- eters, Hi and H 2, respectively : H2 = F(Hi). (24) It is then required to find a mechanism and domain of operation such that Eq. (24) can be transformed into the given Eq. (22) by direct or comple- mentary identification of Hi with hi, with H2 with h2. The usefulness of homogeneous parameters and variables will be abundantly illustrated in the chapters to follow. 3*3. An Operator Formalism. — It is often necessary to combine mechanisms in series, in such a way that the output parameter of the first becomes the input parameter of the second, and so on. The first mechanism determines an output parameter X2 as a function of the input parameter X\i X2 = 4>i(Xi). (25a) The second mechanism determines an output parameter X3 in terms of X2, X3 = 4>2(X2); (256) the third determines an output parameter X4 in terms of X3, X, = <£3(X3); (25c) and so on. The final output parameter, for example, X4, is then deter- mined as a function of Xi: X4 - fo{U<t>i(Xi)]}. (26) The conventional notation of Eqs. (25) and (26) is fully explicit, but some- times cumbersome. For many purposes the author finds it more con- venient and more suggestive to use the following operator notation. 50 BASIC CONCEPTS AND TERMINOLOGY [Sec. 3-3 Equation (25a) implies that the value of X* can be obtained by carry- ing out an operation (of character specified by the definition of <f>i) on the value of Xi. As an alternative notation we shall write X2 = (X,|Xi) • Xx, (27a) where (X2|Xi) denotes an operator converting the parameter Xi into the parameter X2. Similarly, Eqs. (256) and (25c) become X3 = (X3|X2) • X2, (276) X4 = (X4|X3) • X3. (27c) In this notation Eq. (26) becomes X4 = (X4|X3) • (X.|X2) • (X2|X0 • Xt. (28) This form shows clearly the successive operations carried out upon Xi to produce X4. It will be noted, however, that the operators are dis- tinguished from each other only by specification of the parameters involved; it is not possible to change the argument of a given function, as in the conventional functional notation. The over-all effect of Eqs. (27) is to define X4 as a function of Xi: X4 = (X4|Xi) • Xi. (29) On comparing Eqs. (28) and (29) we obtain the operator equation (X4|X.) • (X.|X2) • (X2|Xi) = (X4|Xi). (30) The form of this equation calls our attention to a possible manipulation of these functional operators. In a meaningful product of operators, each internal parameter will occur twice in neighboring positions in adjacent operators. One can, without changing the significance of the operator, strike out such duplicated symbols and condense the notation thus: (X4|X8) • (X.|X2) • (X2|X0 -> (X4|X.) • (X,|Xi) -> (X4|Xi), (31a) or (X4|X8) • (X,|X2) • (X2|X0 -* (X4|X2) • (X«|Xi). (316) Conversely, one can describe the structure of an operator in more detail, with consequent expansion of the notation : (X4|Xi) -> (X4|X8) • (X,|Xi) -4 (X4|X.) • (X8|X2) • (X,|Xi). (32) The inverse operator to (X2|Xi) will be (Xi|X2). Thus Xi - (Xx|X2) • X2, (33) (Xi|Xt) • (X2|X0 ■ 1. (34) Both sides of an operator equation can be multiplied by the same operator. This must be done in such a way that the resulting operators have meaning: the multiplied operators must have neighboring symbols in Sec. 3-4] GRAPHICAL REPRESENTATION OF OPERATORS 51 common. Thus one can multiply both sides of Eq. (30) from the left by the operator (X2|X4), to obtain (X2|X4) • (X4|X3) • (X3|X2) • (X2|X0 = (X2|X4) • (X4|X0, (35) which may be condensed to (X2|X3) • (X3|X0 = (X2|X4) • (X4|XX). (36) Multiplication of Eq. (30) by (X2|X4) from the right is not defined, but multiplication from the right by, for example, (Xi|X3) is defined. This operator formalism can be applied to variables as well as to parameters. An input scale, which determines a parameter X; as a func- tion of a variable xi} can be represented by an operator (Xt|z;) ; an output scale would be represented by an operator (xk\Xk). 3*4. Graphical Representation of Operators. — The operator (X*|Xt), like the function <fc(Xt), is conveniently represented by a plot of Xk against X*. This representation is most uniform and most useful when homogeneous parameters or variables are used. A plot of Hk against Hi always lies in a unit square (Fig. 3-6); it can be used in the graphical construction of curves representing products of the operator (Hk\Hx) with other operators, and in the solution of other types of operator equations, in a way which will now be explained. Given the analytic form of the relations symbolized by Hk = (Hk\Ht) • Hi} (37a) #3 - (H8\Hk) ■ Hk) (376) one can determine the form of the relation H8 = (H8\Hi) • Hi (37c) by eliminating the parameter Hk. In the same way, one can determine the graphical representation of the product operator (H.\Hi) = (Hs{Hk) • (Hk\Hi) (38) by graphical elimination of the parameter Hk from plots of (H8\Hk) and (Hk\Hi). Figure 3-7 illustrates the required construction. The opera- tors (Hk\Hi) and (H8\Hk) are represented, in the standard way, by plot- ting the first parameter vertically against the second horizontally. In the representation of (Hk\Hi), Hk is thus plotted vertically, but in the repre- sentation of (H8\Hk) it is plotted horizontally. The parameter Hi is plotted horizontally in the first case, and H8 vertically in the second ; it is in this way that they are to be plotted in the standard representation of the product operator (H8\Hi), which we must now construct. On the main diagonal of the square, the line (0, 0) — > (1, 1), we select a point A; 52 BASIC CONCEPTS AND TERMINOLOGY [Sec. 3-4 this will represent, by its equal horizontal and vertical coordinates, a par- ticular value of the parameter Hk. A horizontal line through A will intersect the curve (Hk\Hi) at a point B; the horizontal coordinate of B is a value of Hi corresponding to the chosen Hk. A vertical line through A will intersect the curve (Hs\Hk) at a point C; the vertical coordinate of C is the value of Hs corresponding to the chosen Hk. The point D, con- structed by completing the rectangle ABDC, then has the horizontal coordinate Hi and the vertical coordinate Hs corresponding to the same Fig. 3-6. — Graphical representation of typical operator (Hk\Hi). Fig. 3*7. — Construction of a product of operators. value of Hk; it is a point on the curve of the product operator (Hs\Hi). It mil be noted that the horizontal line through A intersects the curve (Hk\Hl) at a second point, B' , to which corresponds a second value of Hi compatible with the same values of Hk and Hs. The point Df deter- mined by constructing the rectangle AB'D'C is thus a second point on the curve (Hs\Hi). By carrying out this construction for a sufficient number of points A, one can determine enough points D, D', on the curve (Hs\Hi) to permit its construction with any desired accuracy. The slopes of the factor and product curves are simply related. The analytic relation dH, dHi dHs dH_k dHk ' dHi (39) becomes, in the notation of Fig. 3-7, [Slope of (H8\Hi) at D] = [Slope of (H8\Hk) at C] X [Slope of (Hk\Hi) at B]. (40) If the factor curves intersect at a point A on the main diagonal, the rectan- gle ABDC reduces to a single point; the product curve passes through this same point, with a slope equal to the slopes of the factor curves. An important special case is that in which both factor functions are con- Sec. 34] GRAPHICAL REPRESENTATION OF OPERATORS 53 tinuous and monotonically increasing in the range of definition. The factor curves then intersect at the points (0, 0) and (1, 1), at the ends of the main diagonal ; the terminal slopes of the product curve are equal to the products of the corresponding terminal slopes of the factor curves. It is sometimes desirable to construct the product (Hs\Hk) • (Hk\Hi), using, instead of a plot of (Hs\Hk), a plot of its inverse, (Hk\H8). The required construction is shown in Fig. 3-8. A horizontal line through a point A , corresponding to an arbitrarily chosen value of H k, will intersect the curve (Hk\Hs) at a point C with horizontal coordinate HS) and the curve (Hk\Hi) at a point B with horizontal coordinate Hi. A vertical line through C will intersect the main diagonal at a point D with vertical coordinate Hs. Finally, by completing the rectangle CDEB, one can Fig. 3-8. — Construction of the product (Hs\Hk) • (Hk\Hi), using plot of (Hk\Hs). Fig. 3-9. — Graphical solution of Z • (Hi\Hk) = (Hs\Hk). determine the point E, with vertical and horizontal coordinates H8 and Hi} respectively; this point, then, lies on the required curve (Hs\Hi). This construction is essentially a solution of the operator equation (Hk\Hs) • (Hs\Hi) = {Hk\Hx), (41) the first and third of these operators being known. Otherwise stated, it is a graphical solution of the operator equation (Hk\Hs) • Y = (Hk\Hi) (42) for the unknown operator Y, which is obviously the desired (Hs\Hi). It will be noted that the construction of Fig. 3-8 is that required for the multiplication of (Hk\Hs) and (Hs\Hi) to produce {Hk\Hi), according to the method first explained. Another operator equation often encountered is Z • (Hi\Hk) = (H3\Hk). (43) 54 BASIC CONCEPTS AND TERMINOLOGY [Sec. 3-5 The construction for Z is sketched in Fig. 3-9 in the case of monotone operators (Hi\Hk) and (H8\Hk). 3-6. The Square and Square-root Operators. — It is sometimes desir- able to connect in series two identical linkages with equal input and out- put travels. The first linkage carries out the transformation Hk = (Ht\Hi) • Hit (44a) the second linkage, the transformation Hs = (H.\Hk) • Hk, (446) where the operators (Hk\Hi) and (Hs\Hk) are identical in form, though not, of course, in the arguments. Then (Ht\Hi) = (Hs\Hk) ■ (flJLHi) (45) Fig. 3*10. — Squaring an operator is essentially the square of the operator w = (Ha\Hk) m (H*|#0; Fig. 3-11. — Squaring an operator W represented by a curve which crosses the main diagonal. Eq. (45) may be written as mm w-w = w\ (46) (47) The construction for the operator W2 is illustrated in Figs. 3-10 and 3-11. In principle, it is the same as the construction of Fig. 3-7; differences in appearance arise from the fact that, since the functions are identical, the points B and C He on the same curve, instead of on two different ones. The curve representing W2 lies beyond the TF-curve, away from the main diagonal. Where the IT-curve crosses the main diagonal, the IT2-curve also crosses it, with a slope equal to the square of the slope of the W-curve ; terminal slopes are related in the same way when the terminal points are (0, 0) or (1, 1). Thus the variations in slope of the IT2-curve, and its curvature, are greater than those of the IF-curve. Sec. 35] THE SQUARE AND SQUARE-ROOT OPERATORS 55 Fig. 3-12. — Construction square-root operator. of The difficulty in designing a linkage to generate a given function tends to increase with the curvature of the function. It is often impossible to use a linkage of given type to mechanize a given functional operator (Hs\Hi) = W2 with large curvature, but quite feasible to mechanize the less strongly curved square-root operator W. If it is possible to solve Eq. (47) for the operator W, and to mechanize this by a linkage with equal input and output travels, it is then possible to mechanize the given func- tion by two such linkages in series. This technique will be discussed in Chap. 6; we shall here consider only the graphical method for solving for the square-root operator W, when W2 increases monotonically. The general nature of the problem of solving for W can be understood by in- spection of Fig. 3-10. One needs to fill out the region between the main diagonal and the TF2-curve by a system of rectangles with horizontal and vertical sides, such that one corner of each rectangle lies on the main diagonal, the opposite corner lies on the IF2-curve, and the other two corners fall on a continuous curve, the W-curve. This can always be done, and in an infinite number of ways; the square- root operator is not unique, but has the multiplicity of the curves that can be drawn between two given points. A square-root operator can be constructed in the following way. Between the main diagonal and the IF2-curve, let a point C be chosen, quite arbitrarily (Fig. 3-12). Beginning at the point C, construct the horizontal fine a/3, the vertical fine (3y, the horizontal line yd, and so on; these form a step structure with vertexes alternately on the main diagonal and the JF2-curve, extending through the region between these lines. A second step structure passing through C is formed by the vertical line £'a'} the horizontal line a'(3', the vertical line (3fyr, and so on. These two step structures define a series of rectangles with opposite vertexes on the main diagonal and the TP2-curve. The other vertexes define a sequence of points, . . . , A, B, C, D, . . . , such that a IF-curve which passes through any point of the sequence, say C, must pass also through all the others. This sequence of points will have a point of condensation where the TF2-curve crosses the main diagonal, and cannot be extended through such a point. In Fig. 3-12 the points of condensation are the terminal points (0, 0) and (1, 1); in a case like that of Fig. 3*11, inde- pendent sequences must be defined in regions separated by points of condensation. 56 BASIC CONCEPTS AND TERMINOLOGY [Sec. 3-5 Let us choose to construct a square-root operator, W, which passes through the sequence of points, . . . , A, B, C, D, . . . , indicated by solid circles in Fig. 3-12. We can also require that it pass through any other similarly constructed series of points, . . . , A', B', C, D', . . . , such as that indicated in Fig. 3-12 by small circles. We can, in fact, com- pletely define W by requiring that it pass between points B and C in an arbitrarily chosen continuous curve. Corresponding to the points of this curve, the above construction will define sequences of points that con- nect A to B, C and D, and so on; these points define a continuous TF-curve extending from one condensation point to the next. The reader will find it easy to prove that if W is to be single-valued everywhere, it must increase monotonically between B and C. / / y« / fpff A / /« / / V b* / Ac*/ A/ /<d* /7/e* Fig. 3-13. — Construction of a squareroot operator near a point of condensation. Fig. 3.14. — Square-root-operator curve having a derivative at a point of conden- sation. The square-root operators thus defined do not, in general, have derivatives at the limiting points of condensation. In Fig. 3-12 it is evident that the TF-curve oscillates more and more rapidly as the origin is approached, and it is hardly to be expected that a derivative will exist at that point. Figure 3*13 represents the part of Fig. 3-12 very near the origin, in a neighborhood in which the TF2-curve can be replaced by a straight fine with finite slope S ^ 1. The points a, b, c, d, e, fall in the same sequence as the points A, B, C, D, of Fig. 3-12. No attempt is made to represent the forms of the intervening curve segments, which are replaced by straight fines. The step structure shown dashed is the con- tinuation of the structure a(3y5e ... of Fig. 3-12; it will be unchanged if the point C is shifted horizontally, say to C*. The other step structure is the continuation of afi'y'b' . . . , and it will be changed by a horizontal shift of C. It is easy to show that the segments ab, cd, ef, . , . , are parallel, as are the segments be, de, fg, . . . . The segments ab and cd are in general not parallel to each other; the average slopes in successive Sec. 3-5] THE SQUARE AND SQUARE-ROOT OPERATORS 57 segments of the TF-curve remain constant and different as the origin is approached, and no derivative exists at the origin. As we have already noted, a shift of the point C of Fig. 3-12 to the left will modify one of the step structures, defining a new sequence of points a*, b*, c*, . . . , corresponding to the new point C*. By proper choice of C* the new sequence of points can be brought to he on a straight line through the origin, as shown in Fig. 3-14. Only through this particular sequence of points can one pass a W-curve having a derivative at the origin; the limiting slope of that curve must be the slope of the line a*b*c* . . . , which is easily shown to be y/S. This geometric argu- ment thus leads to the already stated conclusion that the slope of the W-cmve at a point of condensation must (if it exists) be equal to the square root of the slope of the TF2-curve. The argument of the preceding paragraph also leads to the conclusion that on any given horizontal line there is one and only one point C* that lies on a TF-curve with derivative at the origin. It is evident, then, that the condition that the W-c\xrve shall have a derivative at the origin (or any other point of condensation where the TF2-curve intersects the main diagonal with a finite difference of slope) is sufficient to determine uniquely the form of the TT-curve as far as the next adjacent point of condensation. Since an independently determinable section of the W-curve usually lies between two such points of condensation, the condition that it have a derivative everywhere places on it two conditions, which may or may not be consistent. Thus for any given monotonic TF2-curve there can exist, at most, one TT-curve with a derivative everywhere; there may exist none at all. If the JF-curve is to be mechanized exactly, it is obviously necessary that it have a derivative everywhere. For an approximate mechanization it is only necessary that the TF-curve oscillate with sufficiently small amplitude about a mechanizable curve with a derivative everywhere. In either case, the analysis just outlined forms a practical basis for the determination of IT-curve. Trying in turn several points C, one can quickly find a point C* such that the slopes of the segments a*/?*, 0*y*, . . . approach equality as one of the two limiting points of condensation is approached. The corresponding slopes may then oscillate near the other point of condensation, at which this IT-curve will have no derivative. It is, however, usually possible to choose C* so that the oscillations of the JF-curve are negligibly small near both points of condensation. By inter- polation one can then determine a smooth approximate W-curve suitable for mechanization. CHAPTER 4 HARMONIC TRANSFORMER LINKAGES We turn now to the problem of designing a bar linkage for the mecha- nization of a given functional relation between two variables. The devices used will be discussed in the order of their increasing flexibility and the increasing complexity of the design procedure required: in Chap. 4, harmonic transformers and double harmonic transformers; in Chap. 5, three-bar linkages; in Chap. 6, three-bar linkages in combination with harmonic transformers or other three-bar linkages. Full examples of the design techniques will be provided by detailed discussions of the problem of mechanizing the tangent and logarithmic functions. THE HARMONIC TRANSFORMER 4*1. Definition and Geometry of the Harmonic Transformer. — An ideal harmonic transformer is a mechanical cell for which input and output parameters Xi and Xk are related by Xk = R sin Xi, (1) R being an arbitrary constant. Such a relationship can be obtained with simple mechanisms modeling a right triangle, such as are sketched in Fig. Ideal harmonic transformers. 4-1. These harmonic transformers are called "ideal" because they generate the sine or arc-sine functions accurately ; unfortunately, they are somewhat unsatisfactory mechanically, and are therefore used only exceptionally in practical work. It is usually preferable to employ nonideal harmonic transformers, such as those shown in Figs. 4-2 and 4-3, 58 Sec. 41] DEFINITION OF THE HARMONIC TRANSFORMER 59 which give only an approximately sinusoidal relation between input and output parameters. The mechanism shown in Fig. 4-2 is an ordinary crank-link system with unsymmetrically placed slide. The deviation of the output parameter from its " ideal" value depends upon the angle e between the link L and the line of the slide. Representing the output parameter by X'k, one has Xk = R sin Xi - L(l - cos €). This may be written as X'k — Xk + 5X&, (2) (3) Fig. 4-2. — Crank-link system as a nonideal harmonic transformer. where 8Xk is the structural error of the mechanism as compared with the ideal harmonic transformer: 5Xk = — L(l — cos «). In the mechanism of Fig. 4-2, e is variable, being given by L sin e = R cos X, — W. (4) (5) In Fig. 4-2 the slide displacement W has been so chosen as to keep e, and hence BXk, small as the crank turns through its limited operating angle. As will be discussed in detail later, it may be desirable to make a different choice of W in order to obtain a desired non vanishing form for &Xk. 60 HARMONIC TRANSFORMER LINKAGES [Sec. 4-1 Figure 4-3 represents a harmonic transformer connected to another linkage such that the pivot P may be found anywhere within the shaded area. Equations (2), (3), and (4) hold in this case, but e and the struc- tural-error function 8Xk now depend not only on Xi, but also on the position of the pivot P within the possible boundary. An ideal harmonic transformer generates a section of sine or arc-sine curve, the form of which can be fixed by specification of the angular limits of the rotation of the crank, Xim and XiM . The nonideal harmonic trans- former requires four parameters for its specification — for instance, Xim, XiM, L/R, and W/R. The presence of these additional parameters per- Fig. 4-3. — A nonideal transformer without fixed slide. mits a considerable extension of the field of mechanizable functions — an extension which becomes striking if e is permitted to assume large values. In most practical work e and 8Xk are kept fairly small; 8Xk then appears either as an error arising from the use of a nonideal design, or as a small correction to the sinusoidal form, by which one makes the mechanized function correspond more closely to a given, not exactly sinusoidal, function. In working out the mathematical design of a system that includes a nonideal harmonic transformer, it is usually desirable to carry through the first calculation as though the transformer were ideal. The error arising from use of the nonideal design can then be corrected in the final stages of the work (if this is required by very rigid tolerances), or so chosen as to minimize the over-all error of the system. Sec. 4-2] USE OF THE HARMONIC TRANSFORMER 61 4-2. Mechanization of a Function by a Harmonic Transformer — In the harmonic transformer one parameter is a rotation, the other a translation. Either of these may be taken as the input parameter. If the crank R is the input terminal, the limits of the input parameter X{ may be chosen at will; the crank can describe any angle or make any number of revolutions. The mechanized function will always be a sinusoid or a part of a sinusoid between chosen limits (Fig. 4-4) . If the slide is the input ter- minal, the range of the input parameter Xk must be limited to keep the mecha- *k xJ y/k x* \ V R 'Xkm, x> % XiM A*( Maximum slope Fig. 4-4. — Sinusoid generated by an ideal harmonic transformer. Fig. 4-5. — Arcsinusoid generated by an ideal harmonic transformer. nism far enough from the self-locking positions. The mechanized function is then a portion of an arcsinusoid (Fig. 4-5) within which the slope does not exceed some maximum value determined by mechanical considerations. The simplest problem in ideal-harmonic-transformer design is that of mechanizing a harmonic relation, analytically expressed, between varia- bles Xi and xk: xk — xko = r sin (x{ — xio), r > 0, (6a) or Xk Xko • _i (%k — Xk\ Xi — xio = sin x I J, (6&) given specified limits for the input variables. To determine the constant R of the harmonic transformer and the required relation of the variables x^ Xk, to the parameters Xi} Xk) one need only compare Eqs. (1) and (6) : 5L^£i. = **, Xi-Xia = Xi. (7) The value of R, chosen at will, determines the scale factor Kk of the param eter Xk: xk — xko = xk — xho = _r Xk — Xka Xk R Kk = (8) 62 HARMONIC TRANSFORMER LINKAGES [Sec. 4-3 [Xjk0 = 0, by Eq. (7)]. The scale factor for X, is unity. These constants being fixed, the harmonic transformer is determined. The range of parameter values for which it must operate is determined by the limited range of the input and output variables, xim ^ x{ ^ xiM, Xkm ^ xk ^ xkM'. Xim = Xim — %io, XiM = XiM ~ Xi0, (9) XXkm Xk0 v %kM Xko /irv\ km ~ ■ r? > AkM = t? (,1U) Afc Afc A less trivial problem is that of mechanizing a function that has a generally sinusoidal character, but is given only in tabulated form. One possible method in such a case is to fit the given function as well as possi- ble (for example, using the method of least squares) by the analytic expressions of Eq. (6), and then to proceed as just explained. A quicker way, making use of homogeneous variables and parameters, will now be presented. 4*3. The Ideal Harmonic Transformer in Homogeneous Parameters. Before expressing the equation of an ideal harmonic transformer in homo- geneous parameters, we must define the parameters more precisely. The position of the crank R (Fig. 4-2) is described by the parameter Xi, the rotation of the crank clockwise from a zero position perpendicular to the center line C of the slide. The other parameter, Xk, is defined as the normal projection of the arm R onto the center line of the slide. The crank R in the zero position is pictured as directed upwards, and Xk is taken as positive toward the right from the point S. The homogeneous parameters 0», Hk, are related to the parameters Xi, Xk, by /j Xi A,m jj X.k Afcm /-ii\ $i AX~' Hk = AXk ' (11) (The symbol 0; is chosen to represent one homogeneous parameter, instead of Hif to emphasize the fact that in this case one is concerned with a rotation.) From these definitions it follows that both homo- geneous parameters increase in the same sense as the original parameters : 6i increases always clockwise, Hk increases to the right. The connection between ordinary and homogeneous parameters in a harmonic transformer is illustrated in Fig. 4-6. The arc of the angle of travel AXit scaled evenly clockwise from 0 to 1, permits direct reading of 0». The projection of that arc on a straight line perpendicular to the zero line SO, scaled evenly from 0 to 1, from left to right, permits direct reading of Hk. Any line parallel to OS passes through corresponding values of 0» and Hk. The correlation of values of Hk to those of 0i is unique so long as AX» < 360°; the converse correlation may be double- valued in some cases, as is illustrated by Figs. 4-6(6) and 4-6(c). Sec. 44] TABLES OF HARMONIC-TRANSFORMER FUNCTIONS 63 From the definition of homogeneous parameters and from Eq. (11) it is evident that, always, Hk = sin (Xim + OiAXi) - (sin Xj\ (sin Xi), (sin Xi)t (12) XiM Fig. 4-6. — Ideal harmonic transformer with homogeneous parameters, (a) (sin XtOniHx = sin XiM. (&) (sin X;)max =1. (c) (sin Xt)min = sin XiM. Special forms of this relation, applicable in cases of the types illustrated in Figs. 4-6(a), 4-6(6), 4-6(c), respectively, are as follows: „ sin (Xim + OiAXi) - sinX, Hk • v : v sin XiM — sin Xim „ sin (Xim + OiAXi) - sin X Hk = Hk = 1 — sin Xim sin (Xim + OiAXi) - sin Xt- 1 — sin Xi (13a) (136) (13c) 44. Tables of Harmonic -transformer Functions. — The use of har- monic transformers as parts of complex linkages is so extensive and the design problem is so greatly simplified by the use of homogeneous param- eters that it is very convenient to have available a fairly complete table of the functions appearing in Eq. (13). Table A-l gives Hk for Si = 0.0, 0.1, 0.2, • • • 0.9, 1.0, and for AXt- = 40°, 50°, • ■ ■ 140°. Smaller values of AX; are of little interest, since with small angular travel the errors due to mechanical play become relatively important, and other devices can serve as well for mechanization of the corresponding nearly linear functions Hk(0i). Two facing pages are required for each value of 64 HARMONIC TRANSFORMER LINKAGES [Sec. 44 AXi. Columns of values of Hk are grouped in pairs in a way intended to facilitate the calculation of structural error functions, as discussed in Sec. 4-5. The first of these columns has the corresponding values of Xim and XiM indicated at the top, and is tabulated with 0f (indicated to the left) increasing downward. The second column has the values of Xim and XiM indicated at the bottom, and is tabulated with 0* (indicated at the right) increasing upward. The associated columns correspond to har- monic transformers with Xim ^ Xi ^ XiM and with (90° - XiM) ^ Xi ^ (90° - Xi,.), respectively; the significance of this and of other features of the table which are not of importance at this point will be explained in Sec. 4-5. Table A-2 gives 0; for Hk = 0.0, 0.1, 0.2, • • • 0.9, 1.0, and for the same AXi as Table A-l. The arrangement is simple and should require no explanation here. Only single-valued relationships between Hk and 0i are tabulated, since the table is intended for use when Hk is the input variable; all regions that include a point with infinite ddi/dHk may be excluded. In using Tables A-l and A-2 to mechanize a tabulated function with a pronounced sinusoidal character, the function should first be expressed in homogeneous variables. We shall call the homogeneous input variable hr, the homogeneous output variable h8, in order to avoid any commitment as to which is to be the angular parameter in the mechanization. Next, there should be tabulated in a column the values of the output variable h8, for hr = 0.0, 0.1, 0.2, • • • 1.0, making such interpolations as may be necessary. It remains only to compare this column of numbers with those in Tables A-l and A-2. One can easily find which of these columns gives the best fit to the given set of numbers; each column, it is important to note, may be read either up or down. This determines the best values of Xim and XiM for the harmonic transformer, to within 10° ; by interpolation one may fix these values even more precisely. The remainder of the design process is then trivial. If the best fit is found in Table A-l, the output variable h8 is being identified with Hk; the output terminal of the mechanization will be the slide, the input terminal the crank. If the best fit is found in Table A-2, the reverse is true. Suppose that the best fit is found in Table A-l, and that, in reading the corresponding columns, hr and 0* increase together. Then one has hr = 0», hs = Hk. Knowing Xim and XiM, one can construct scales of 0» and Hk as described in the preceding section; these are the required scales of hr and h8, which one can recalibrate in terms of the original variables, if this should be desired. Sec. 4-4] TABLES OF HARMONIC-TRANSFORMER FUNCTIONS 65 If the best fit is found in Table A-l, but correspondence of the columns requires that they be read in such directions that hr decreases as 0; increases, then 1 — hr = Bi, h8 = Hk. The /ir-scale thus differs from the 0t-scale only in that hr increases to the left instead of the right; the rest of the construction is as before. If the best fit is found in Table A-2, one has h8 = Bi, hr = Hk, if h8 and Bi increase together, and otherwise 1 — h8 = Bi, hr = Hk. In the operational language introduced in Chap. 3 this process may be described as follows: A functional operator Qi8\hr) is given, and there is sought a functional operator (Hk\0i) or (6i\Hk) of a harmonic transformer which transforms into the given operator (h8\hr) when the pair of variables (hr, h8) is transformed into the pair of parameters (0t, Hk) or (Hk, 0») through a direct or complementary identification. When the tables are employed it is useful to make graphs of operators and sketches of mechanisms in order to prevent mistakes. It is recom- mended that the Hk-sc&le run always from left to right, that the zero line for Xi be directed upward, and that the scale for 0* increase clockwise, as in Fig. 4-6. Example: Use an ideal harmonic transformer to mechanize the relation x2 = tan xi (14) with the range of the input variable X\ from 0° to 50°. The homogeneous variables are hr = ^, h. = ^|^- (15) Table 4-1 gives the relation of h8 to hr in tabular form. Table 4-1. — x2 = tan Xi, 0 ^ Xi ^ 50°, in Homogeneous Variables K h, 0.0 0.0000 0.1 0.0734 0.2 0.1480 0.3 0.2248 0.4 0.3054 0.5 0.3913 0.6 0.4844 0.7 0.5875 0.8 0.7041 0.9 0.8391 1.0 1.0000 In seeking a corresponding column in the tables, we need examine only those which show no maximum. In such cases the first and last values are always 0 and 1 ; every such column matches the given column at the two ends. 66 HARMONIC TRANSFORMER LINKAGES [Sec. 44 Consider first Table A-l. Fixing on a value of AX,-, we seek a column that gives a match at the middle as well as at the ends; for example, with AXi = 70° the best match is obtained for Xim = -70°, XiM = 0°. However, this column contains values that are too small at small 0», too large at large 0t. Repeating this process for smaller AX{, one obtains a better over-all fit, but the improvement is slight; one must either use very small values of AX* or tolerate errors of over 2 per cent of the total range. Next we examine Table A-2. Again the best match is obtained for relatively small AXt — a consequence of the nearly linear character of the tangent function in the given range. Here, however, a much better match is possible. Comparing with the given ha the values of 0; shown in Table A-2 for Xim = 30°, XiM = 70° and for Xim = 35°, XiM = 75°, one finds the differences shown in Table 4-2. Table 4-2. — Values of 7is-0» Hk Xim = 30° Xim = 35° Xim = 31.5° XiM = 70° XiM = 75° XiM = 71.5° 0.0 0.0000 0.0000 0.0000 0.1 -0.0004 0.0036 0.0008 0.2 -0.0023 0.0056 0.0001 0.3 -0.0050 0.0065 -0.0015 0.4 -0.0076 0.0072 -0.0032 0.5 -0.0097 0.0080 -0.0044 0.6 -0.0106 0.0095 -0.0046 0.7 -0.0095 0.0120 -0.0030 0.8 -0.0060 0.0152 0.0003 0.9 -0.0009 0.0160 0.0042 1.0 0.0000 0.0000 0.0000 Linear interpolation between these columns shows that with Xim = 31.5°, XiM = 71.5° the difference between hs and 0; remains less than 0.005; an ideal harmonic transformer with these constants would have a structural error everywhere less than 0.5 per cent of the travel. Figure 4-7 shows the harmonic transformer thus designed, with func- tional scales for ha = Hk and hr = 0*. The travel, AXk, can be given any desired value by proper choice of R : AXk = R(sm 71.5° - sin 31.5°) = 0.425872. (16) It is interesting to note that in this example the angular variable X\ of Eq. (14) has been mechanized as a slide displacement, the linear variable x2 as an angular displacement, whereas in a constructive computer the reverse would be the case. Sec. 4-5] TOTAL STRUCTURAL ERROR 67 In this design procedure we have treated the harmonic transformer as ideal. To construct it as nonideal would introduce an additional struc- tural error, 8Xk, described by Eq. (4) — an error that can be made suf- ficiently small by making the link L very long and by so placing the center line of the slide as to reduce the maximum value of the angle e as much as possible. In general it is better to make positive use of the term 5Xk, so choosing the design constants that 8Xk tends to cancel out the structural error 8hk of the ideal-harmonic-transformer component of the mechanism. In the present case this may seem hardly worth the trouble, as the fit obtained with the ideal transformer is very good. However, it is to be Fig. 4-7.- -Harmonic transformer mechanizing X2 = tan Xi, design is shown in Fig. 4-12. 0° < Xi < 50°. A better noted that this design is unsatisfactory in that the angular travel AXt- is rather small. In practice, it would be better to employ a nonideal trans- former with large angular travel, keeping the total structural error small by judicious choice of L and the slide position (Fig. 4-12). The required design technique is discussed in the next sections. 4-5. Total Structural Error of a Nonideal Harmonic Transformer. — In finding a harmonic transformer to mechanize a given relation, fa = (hk\hi) • hi, (17) one begins, as already described, by finding an ideal harmonic transformer that gives an approximate fit. Then if 0f is identified with hi} Hk can also be identified with hk, except for the small structural error 8hk: Hk = hk + 8hk. (18) 68 HARMONIC TRANSFORMER LINKAGES [Sec. 4-6 If the transformer to be used is nonideal, its output parameter will be not Hk but Hk. Representing by 8Hk the change in output arising from the nonideal character of the transformer, we write H'k = Hk + 8Hk. (19) The complete mechanism then has a structural-error function 6h'k = H'k-hk = Mk + dhk; (20) it is this error that should be reduced to tolerable limits over the whole range of operation. A nonideal harmonic transformer has been sketched in Fig. 4-2. Of the four design constants, Xim and XiM characterize the ideal-harmonic- transformer component and determine the form of 8hk; L/R and W/R affect only the form of 8Hk. It is of course impossible in general to make 8hk vanish identically by any choice of these parameters. Ideally, one would manipulate all four parameters in order to make 8hk every- where satisfactorily small, without regard to the resulting magnitude of 8Hk and 8hk. An easier technique is to make 8hk as small as possible by choice of Xim and XiM , and then to choose L/R and W/R so as to minimize 8hk; however, one can often arrive at more satisfactory designs, and even appreciably reduce the over-all error, by some other choice of Xim and X{M. 4*6. Calculation of the Structural-error Function 8Hk of a Nonideal Harmonic Transformer. — In designing harmonic transformers it is impor- tant to have a quick, efficient way to compute the structural-error func- tion 5Hk. Use of Eq. (4) is neither quick nor well adapted for work with homogeneous parameters; better methods to be described here and in Sec. 4-7 depend upon reference to Table A-l. The discussion will be illustrated by Fig. 4-8, which shows a harmonic transformer with alterna- tive positions for the link L, extending from the crank toward the left or toward the right. Here, and throughout the discussion that follows, the unit of length, in which all dimensions are stated, is taken to be the length of the i/^-scale; thus, AXk =1. As before, we consider the harmonic transformer in its basic position, with 9i increasing clockwise, the zero for Xi vertically upward, and scales Hk increasing from left to right. The change from the ideal harmonic transformer (scale Hk) to the nonideal one (scale H'k) will be traced through two steps. First, the Hk-scale may be shifted bodily to the right or left by a dis- tance L. On this scale, shown in Fig. 4-8 above the slide, the reading opposite the pointer will be Hk, modified by an error DHk = ±[L|(1 - cose). (21) Sec. 4-6] CALCULATION OF THE STRUCTURAL ERROR 69 The sign of this error depends only on whether the crank extends to the right or to the left. Taking L as positive when the link extends toward the left, negative when it extends toward the right, one has always DHk = 1,(1 - cose). (22) As Hk changes from zero to one, Hk + DHk changes between limits which are in general not zero and one: (H* + DHk)min ^Hk + DHk ^ (Hk + DHk)mAX; (23) thus Hk + DHk is not in general a homogeneous parameter. As the second step, the Hk + DHk-scale is replaced by the homo- genized Hk scale, shown in Fig. 4-8 below the slide: Hi = Hk + DHk - (Hk + DHk)min (Hk + DH &)max - (Hk + DHk\ (24) i i ■ ■ i i i ■ H{ 0.0 0.5 Hh~DHM 1.0 Hi Fig. 4-8. — Notation used in harmonic transformer design. When T>Hk is reasonably small the maximum value of H k + DHk will occur at essentially the same 0» as the maximum of Hk — that is, when Hk = 1. One can then write (Hm + DH*)— « 1 + (DHk). and similarly (Hk + DHk\ (DHk)0, (25) (26) (DHk)o and (DHk)i being the values of DHk for Hk equal to 0 and 1 respectively. As an approximation good enough for all preliminary calculations one has then Hi Hk + DHh - (DHt), 1 + (DH,), - (DHh)0 (27) To compute DHk, we observe that if Yk is the distance above the slide center line of the pivot between L and R, then Yk sm€ = T (28) 70 and When € is small HARMONIC TRANSFORMER LINKAGES DHk = l[i-(i -£)"]. [Sec. 4-6 (29) V2 (30) The quantity Yk is conveniently found as a function of 0< by use of Table A-l, by taking advantage of the special relationship of associated columns of that table. The relationship of the corresponding harmonic trans- Fig. 4-9. — Harmonic transformers associated in Table A4. formers is illustrated in Fig. 4-9. The first transformer (parameters Xi, Xk'} Oi, Hk) operates through the range Xim ^ Xi ^ XiM (Xim = — 15°, XiM = 75° in Fig. 4-9a). The second transformer (param- eters Xf, X*; 6*, H*) operates through the range XL = 90° - XiM ^ X* ^ X* = 90< Xi (Xfm = 15°, X*M = 105° in Fig. 4-96). Now it will be observed that if Fig. 4-96 is reflected in the line X? = 45° and superimposed on Fig. 4-9a, the angular scales will then coincide, but with 0* = 1 — 6i. The H*-sca\e, however, becomes a vertical scale, as compared with the hori- zontal #*-scale. The entries in a section of Table A-l may then be interpreted as follows: For a harmonic transformer with limits on Xi as given at the top, the four entries in each row, from left to right, correspond to (1) Bi for some index point P on the angular scale, (2) Hk, which meas- ures the horizontal displacement of that point to the right of a vertical Sec. 4-7] STUDY OF THE STRUCTURAL-ERROR FUNCTION 71 reference line, (3) H*, which measures the displacement of the same point upward from a horizontal reference line (though in different units, since the #t-scale is not in general of unit length), and (4) 0? = 1 — 0im If the limits on Xi are those given at the bottom of the section, the entries in each row have the same meaning if they are taken in order from right to left. The quantity Hk{ = -X*) measures the actual distance of the point P from the vertical reference line, since the Hk-sca\e is one unit long by definition. The length of the vertical scale is _ (COS Xj)mai — (COS Xj)mm /oi\ 9 (sin X,)— - (sin X,)min ; {6l) hence the actual distance of the point P from the horizontal reference line is X*k - gHt. (32) Values of g are given in Table A-l, in the same line with those of X»m and XiM and in the same column with the values of H J. Returning now to the computation of Yk, we define E* as the value on the H J-scale at the point where the slide center line intersects it. In Fig. 4-8, E* lies on the calibrated part of the scale. This is not necessarily so; E* is a parameter in the design which may be assigned negative values, or values greater than one. In any case Yi = g(Ht - Et). (33) It is convenient to specify a nonideal harmonic transformer by giving Xim, XiM, E%, and L. Calculation of its structural-error function for a series of values of 0* or Hk then requires reading from Table A-l the cor- responding values of #*, followed by computation of Yfk by Eq. (33), DHk by Eq. (29) or (30) (according to the accuracy required), H'k by Eq. (27), and finally dHk by Eq. (19). An illustrative calculation will be found in Table 4-5. This procedure is quick and easy if E* and L are known, but when it is desired to determine the approximate form of 8Hk for a considerable series of values of E* and L, or to find required values of E% and L, the method to be described in the next section is to be preferred. 4*7. A Study of the Structural-error Function 5Hk. — For a general investigation of the structural-error function 8Hk or for a preliminary (and usually final) choice of E* and L in the process of designing a nonideal harmonic transformer, it is sufficiently accurate to use Eq. (30) in com- puting DHk, and to assume that \DHk - (DHk)0\ « 1. (34) To this approximation 8Hk has a simple dependence on E% and L which facilitates its computation for a series of values of these parameters, or, 72 HARMONIC TRANSFORMER LINKAGES [Sec. 4-7 conversely, the finding^ of values of E% and L which give 8Hk a desired form and magnitude. Expanding H'k, as given by Eq. (27), in powers of the small quantity (DHk)i — (DHk)o, and neglecting terms of the second order of smallness, one finds H'k~Hk + [DHk - (DHk)0] + Hk[{DHk)» - (Dff*)J (35) and mk « [DHk - (DHk)Q] + Hk[(DHk)0 - (Z>ff*)i]. (36) This approximation to 8Hk, like the function itself, vanishes when H k = 0 or 1. By Eqs. (30) and (33), DHk « g(Hf - E*k)\ (37) When this is introduced into Eq. (36) the quadratic terms in E* cancel, and one finds «ff*»|J[/i(ft) + £*72(»i)L (38) where /i(0{) and/2(0;) depend only on the parameters Xim and Jiif of the harmonic transformer. With (#*)0 and (HZ) i the values of #* when Hk has the values 0 and 1, respectively, M0t) = Hr - (fff )5 + EifPt)\ ~ (Hj»a (39) M0t) = -2{H! - (Ht)o + HJLWt). ~ (Hf)!]}. (40) Knowing the form of /i(0<) and fz(6t), one can easily compute 5Hk for a large series of values of L and E*. To this approximation the magnitude of the structural-error function varies inversely with L, but its form is determined entirely by E*. The possible range in forms is easily investigated by computing 8Hk for some value of Ek — for example, for E* = 0, in which case one has simply the first term of Eq. (38) — and then adding to this the function /2(0t) in dif- ferent proportions. Although /2(^i) is easily computed by Eq. (40), it is worth while to take note of its simple analytic form. As functions of Xi} one has rr sin Xj — (sin Xj)^ . . a k (sin X0»~ - (sm X^' { } = cos X, - (cos X<U m (COS At)nu« — (COS Aijnun Let Xa and Xb be the values of Xi for which sin Xi has its minimum and its maximum values, respectively. (These are not necessarily Xim and XiM, nor are they always the angles at which cos Xi has its minimum or STUDY OF THE STRUCTURAL-ERROR FUNCTION 73 Sec. 4-7] maximum values.) Then on combining Eqs. (40), (41), (42), one finds, after some trigonometric manipulation, that Xa + Xb 2 sec /• = (&$s) (cos Xi), (cos Xi)i X L(*-S^-^S} /2 is thus symmetric about the value of 0< corresponding to Xf (43) Xa + X& midway between the values of d{ for which Hk = 0 and #& = 1 ; it is of the form of a sinusoid minus a constant, and vanishes for Hk = 0 and Fig. 4-10. — Structural-error functions for nonideal harmonic transformers. The functions shown are (a) /2(0t), and (6) to (/i) (2L/g2)8Hk for a series of values of Ek*, when Xim = -15°, XiM = 75°. if* = 1. Its general form is thus easily sketched without reference to Table A-l. When Hk increases monotonically with Bt, /2(0») is sym- metrical about di = -J* a fact which makes computation even simpler. To illustrate the change in form of 5Hk with changing E* let us consider the special case of a harmonic transformer for which —15° < Xi < 75°. The variation of Hk with 0* for this transformer is shown by the middle curve of Fig. 4-11. Figure 4-10 shows the form of /2(0»), and of or (44) for a series of values of EJ. When £J is less than —0.5 or greater than 1.5, &Hk has nearly the same form as/2(0i), which is symmetrical about 74 HARMONIC TRANSFORMER LINKAGES [Sec. 4-7 $t = 0.5. To produced desired form of H'h that differs from the given Hk by a symmetrical correction 8Hk, one would thus choose Ek < — 0.5 or Ek > 1.5; to raise the Hk curves in the center one would use a positive L (link to the left) in the first case and a negative L in the second, whereas to depress the curve in the center these orientations of the link would be reversed. To lift or depress the Hk-curves for small di} with little change for 0,- near 1, Ef = 0 is an appropriate choice; to make a change near 0* = 1 but not near d* = 0, one should take Et « 0.75. With El ranging from 0.25 to 0.5 it is possible to depress one side of the curve while raising the other, and so on. These observations of course apply only to the particular harmonic transformer here considered; similar sketches would need to be made as the basis for a discussion of other cases. The magnitude of 8Hk is directly controlled by the choice of L. It will be noted however, that when (2L/g2)8Hk is small, as for E* « 0.5, a particularly small value of L may be required in order to give 8Hk a desired magnitude. In general, it is relatively difficult to depress one side of the curve Hk(6i) while raising the other, and one may find that an impractically small value of L is required to produce a desired effect. On the other hand, if one de- sires merely to reduce 8Hk below some established tolerance one can with advantage make E* « 0.5, since con- veniently small values of L are then acceptable. The magnitude of 8Hk in typical cases is illustrated by Fig. 4-11, in which 8Hk is given for three values of (b) Et- = 0.5 ^~ ^y^ /// /// \fh ' e0 ' / s\ /' yy-wi^d L-z-rjZ -L=+2 /// /// /// /// /// F (c)Et = 1.0 Fig. 4»11. — Hk(0i) for nonideal har monic transformers. Xim = —15° XiM = 75°, L ± 2, Ek* as indicated. Sec. 4-8] DESIGN OF NONIDEAL HARMONIC TRANSFORMERS 75 E%(0.0, 0.5, and 1.0) with L — ±2. The difference between the exact calculations on which these graphs are based and approximate calcula- tions using the results of Fig. 4-10 would not be evident to the eye. It is to be emphasized, however, that final calculations should be made using the exact formulas in all cases in which e approaches 45° (a value which, for mechanical reasons, ought never to be much exceeded). 4*8. A Method for the Design of Nonideal Harmonic Transformers. — The experienced designer of nonideal harmonic transformers will find it possible to guess satisfactorily the required values of E* and L, guided only by visual comparison of the //^-curves with the desired form of H'k, and perhaps a few exploratory computations. On the other hand, a simple and straightforward design procedure can be based on the results of the preceding section. To illustrate this, we return to the problem (Sec. 4-4) of using a harmonic transformer to mechanize the relation x2 = tan xi} for 0° < xi < 50°. Here, however, we shall add the requirement that the angular travel of the transformer shall be twice as great as that previously used: AX> = 80°. Despite the imposition of this additional condition, it remains true that it is best to mechanize X\ as a linear displacement, x2 as an angular displacement: the best fit for Table 4-1 is to be found in Table A-2, rather than in Table A-l. Since Table A-l is to be used in the determination of E% and L it is convenient to retabulate the relation of the homogeneous variables hr and hs for equally spaced values of the variable hs, which is to be identified with 0t-. The result is shown in the first two columns of Table 4-3. The best fit for the relation thus expressed is to be found in Table A-l, for Xt 5°, XiM = 75° — the same values, of course, for which one finds in Table A-2 the best fit to Table 4-1. The fit could be improved somewhat by interpolation in the tables, the best value of Xim lying between — 10° and — 5°. We shall not bother with this interpola- Table 4-3.- —Computations in Designing a Harmonic Transformer K = *» K Ih K - Ih /i /■ (^fc)approx. \e/approx. Wexact 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1359 0.1325 0.0034 0.1378 -0.2701 0.0048 0.0014 0.0018 0.2 0.2683 0.2640 0.0043 0.2225 -0.4858 0.0051 0.0008 0.0010 0.3 0.3934 0.3919 0.0015 0.2565 -0.6433 0.0015 0.0000 0.0000 0.4 0.5097 0.5139 -0.0042 0.2465 -0.7387 -0.0049 -0.0007 -0.0012 0.5 0.6158 0.6274 -0.0116 0.2027 -0.7708 -0.0127 -0.0011 -0.0018 0.6 0.7115 0.7304 -0.0189 0.1396 -0.7387 -0.0197 -0.0008 -0.0018 0.7 0.7967 0.8207 -0.0240 0.0724 -0.6433 -0.0240 0.0000 -0.0010 0.8 0.8726 0.8967 -0.0241 0.0173 -0.4858 -0.0233 0.0008 -0.0002 0.9 0.9401 0.9569 -0.0168 -0.0110 -0.2701 -0.0158 0.0010 0.0002 1.0 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 76 HARMONIC TRANSFORMER LINKAGES [Sec. 4-8 tion, but shall choose Xim = — 5°, XiM = 75°, and throw the entire bur- den of correcting our design on the choice of E* and L. The values of Hk read from Table A-l are shown in Column 3 of Table 4-3. The desired value of 8Hk is then hr — Hk, shown in Column 4 of this table. As the next step, /i(0;) and fz(6i) are computed (Columns 5 and 6). By Eq. (38) we can express 8Hk in terms of these functions: 8Hk = aftfi) + bMdi), (45) where a - £ (46) and 2L 9*E\ b = ^ (47) Our problem is then to make a linear combination of Columns 5 and 6 that will approximate Column 4 as well as possible. It is a simple matter to find the best fit in the sense of least rms error, but an even simpler method will suffice: we shall fit 8Hk to hr — Hk exactly at two chosen points. In applying such a method some discretion is necessary as a poor choice of these points may lead to a bad over-all fit. We choose to make the fit exact at 0< = 0.3 and at 0; = 0.7, assuring a proper height for the principal maximum in 5Hk and a change in sign near the correct value of 0». The error in the mechanization will then vanish for four nearly equally spaced values of 0*: 0.0, 0.3, 0.7, 1.0. We require then 0.2565a - 0.64336 = 0.0015, I >. J (48) Hence a = 0.1385, b = 0.0529. (49) By Eqs. (46) and (47), E* = - = 0.382, ;2 (50) L = #- = 1.788. 2a The corresponding values of 8Hk (as computed by this approximate method) appear in Column 7 of Table 4-3, and values of e = dHk - (hr - Hk), the residual error in the mechanization, in Column 8. The maximum error in the mathematical design thus appears to be about 0.1 per cent of the total travel. The maximum value of sin e for this design is f • n 0(1 - 0.382) „0,0 ,K1* (sin c)^ = yK 788 '- = 0.243, (51) Sec. 4-9] TWO IDEAL HARMONIC TRANSFORMERS IN SERIES 77 a sufficiently small value to assure good accuracy of the approximate formulas employed. Exact calculation of the total design error in the mechanization (last column of Table 4-3) shows that it nowhere exceeds 0.2 per cent, a highly satisfactory result. The device itself is sketched in Fig. 4-12. If excessively large values of e occur in a design thus determined, the exact values of 8Hk will not be in satisfactory agreement with hr — Hk. A further correction in 8Hk is then necessary. This may be added to the original values of hr — Hk, and the process of determining E% and L car- ried through as before. The quantities 8Hk, computed with the resulting constants by the exact formula, should now show better agreement with the desired values (the original hr — Hk) . Repetition of this process will usually lead to a satisfactory design, except when excessively large values Fig. 4-12. — Harmonic transformer mechanizing xi = tan xi, 0° < xi < 50°. of e are called for. In such cases another choice of Xim and XiM may help, or another type of linkage may be required. HARMONIC TRANSFORMERS IN SERIES 4*9. Two Ideal Harmonic Transformers in Series. — With a single harmonic transformer one can mechanize only a relatively narrow field of functions. These devices have also a mechanical disadvantage in that one terminal rotates or is rotated by a shaft, while the other pushes or is pushed by a slide; usually one desires that all cells in a computer have terminal motions of the same type. As a first step in the extension of the field of mechanical functions we consider the combination of two ideal harmonic transformers into an 1 'ideal double harmonic transformer," as shown in Fig. 4-13. This mechanical cell has satisfactory mechanical properties, with both ter- minals moving in straight lines. The field of functions that it can gener- ate can be described by three independent parameters — for instance, by 78 HARMONIC TRANSFORMER LINKAGES [Sec. 4-9 AXi, Xim, Xjm, where AX{ is the range of angular motion common to both arms of the rotating member, and Xim and Xjm are the minimum values for the angular parameters Xi and Xj, which describe the orientation of the two arms. Although a considerable variety in form of the generated function is obtainable by proper choice of these parameters, the ideal double harmonic transformer is best suited to the mechanization of monotonic functions with a mild change in curvature (as in Fig. 4-14) and functions of roughly sinusoidal character (as in Fig. 4-16). Fig. 4-13. — Ideal double harmonic transformer. Mechanically, the action of the double harmonic transformer may be thus described: The input parameter Xi is transformed into a rotary out- put parameter X3 by the first harmonic transformer; this rotation is imparted to a second harmonic transformer, for which it serves as a rotary input parameter, XA; X4 is transformed by the second harmonic transformer into the final output parameter X2. Symbolically, in terms, of the corresponding homogeneous variables, 03 = (03|# 1) • Hlt 04 = (04|03) ' 03 = 03, #2 - (#2|04) ' 04, (52) (53) (54) or, combining these relations, H2 = (tf2|04) • (04|03) • (03|# 1) ' #1 = (#2|03) • (03|# 1) • Hi. (55) Sec. 4-9] TWO IDEAL HARMONIC TRANSFORMERS IN SERIES 79 Fig. 4-14. — Graphical construction of the function generated by a double harmonic trans- former (Fig. 4-13). 0 0.5 1.0 Fig. 4-15. — Ideal double harmonic transformer. 80 HARMONIC TRANSFORMER LINKAGES [Sec. 4-10 From this symbolic equation it is evident that one can find the operator for a double harmonic transformer, (H2\H1) = (Ht\0s) ■ (08|#i) (56) by the graphical multiplication of operators for the component harmonic transformers, as explained in Chap. 3. The operator (6z\Hi) may be obtained from Table A-2, the operator (i/2^3) from Table A-l; they must of course correspond to the same value of AX2. As an example we take a double harmonic transformer (Fig. 4-13) for which -75° ^ X3 ^ 15°; -25° ^ Z4 ^ 65°; AX3 = AX4 = 90°. We find in Tables A-l and A-2 the following relations: Hi 6> 0.0 0.0000 0.1 0 . 1942 0.2 0.3207 0.3 0.4249 0.4 0.5175 0.5 0.6033 0.6 0.6849 0.7 0.7641 0.8 0.8422 0.9 0.9204 1.0 1.0000 04 H2 0.0 0.0000 0.1 0.1106 0.2 0.2263 0.3 0.3443 0.4 0.4616 0.5 0.5754 0.6 0.6828 0.7 0.7813 0.8 0.8684 0.9 0.9419 1.0 1.0000 Figure 4-14 shows graphs of these two operators, and the geometric con- struction required for their multiplication as required by Eq. (56) . The graphical representation of the product (H*\Hi) is an almost circular arc, quite different from the functions mechanizable by a single harmonic transformer. Another typical example of two harmonic transformers in series is shown in Fig. 4-15. The travels are AXZ = AX4 = 90°, with -75° ^ Xi ^ 15°, 45° ^ X2 ^ 135°. The operator (dz\Hi) is the one used in the preceding example, and the operator (Ht\$i) will be found in Table A-l. These operators are plotted and their graphical multiplication indicated in Fig. 4-16. The resulting operator is represented by a deformed sinusoid with its maximum dis- placed to the left. 4-10. Mechanization of a Given Function by an Ideal Double Har- monic Transformer. — As the first step in mechanizing a functional rela- tion by an ideal double harmonic transformer, it should, as usual, be expressed in homogeneous variables: h2 = W/ii) ■ K (57) with hi the input variable, h2 the output variable. One then desires to find ideal-harmonic-transformer operators (i/2^4) and (08|#i) which Sec. 410] MECHANIZATION OF A GIVEN FUNCTION correspond to the same value of AXi and which make (H2\H1) = (H2\dz) • (08|#i) 81 (56) approximate as well as possible to the given operator (hz\hi). It is necessary for mechanical reasons, which apply whenever the slide terminal of a harmonic transformer is used as the input, that (6% Hi) not Fig. 4-16. — Graphical construction of the function generated by a double harmonic trans- former (Fig. 4-15). rlf) involve an infinity in -ttt'i we need consider only those cases for which dtL i Table A-2 is constructed, with -90° < Xim, XiM < 90°. Solution of this problem falls into two steps: 1. A preliminary solution of the problem, by which an appropriate value of AXi is fixed upon and a preliminary choice of XZm and X4m is made. 2. Improvement of the choice of XZm and X4m by a process of succes- sive approximations. 82 HARMONIC TRANSFORMER LINKAGES [Sec. 411 To gain a preliminary estimate of an appropriate value of AXi one may fit the given curve very roughly by a section of sinusoid (by reference to Tables A-l and A-2, or even by a visual estimate) ; the angular range of this section of sinusoid will be approximately the desired value of AXi. The roughness of the approximation will be evident from inspection of Figs. 4-14 and 4-16, in both of which the curves correspond to AXi = 90°. However, the nature of the calculations required in computing double- harmonic-transformer functions is such that it is desirable to begin an attempt to fit a given function by fixing on a value AXi, even when the choice must be made quite arbitrarily. By adjusting the parameters X3m and X4m one can then, in principle, obtain the best fit of the mecha- nized function to the given function consistent with the chosen AX»; by repeating this for a series of values of AXi one could at length determine the best value of this parameter and the best possible fit to the given function. In practice, it is not necessary to find the best fit carefully for each AXi. In the preliminary calculations it is sufficient to use a simple and easily applied method of fit in choosing X3m and X4m, to establish an equally simple criterion for the accuracy of the over-all fit thus obtained, and to choose the best AX; in the sense of this criterion. When a value of AXi has been established in this way, it then becomes worth while to use more careful methods, described in Sec. 4-13, in the further adjust- ment of Xzm and X4m. We shall consider separately the quite different methods of getting a preliminary fit to monotonic functions (Sec. 4-11) and to functions with maxima and minima (Sec. 4-12). 4*11. Preliminary Fit to a Monotonic Function. — A monotonic func- tion will in general be fitted by a monotonic function ; the range of X4 will not include either +90° or —90°. In this case one has automatically a fit of the generated function to the given function at both ends of the range of variables. In addition, for any given AXi the values of X3m and X4m can be so chosen that the generated function will (1) agree with the given function at any chosen pair of interior points, or (2) have the same slopes as the given function at the two ends of the range of the input variable, or (3) have the same ratios between the slopes at any three points in the range of the input variables. The first of these methods of fitting would in many cases be the most satisfactory; however, it is the most difficult to apply and will not be considered further. The second method has some- what wider utility than the third and will be made the basis of our further discussion. When X3m and X4m are so chosen that the generated function not only fits the given function at the end points but has the same slope as well, a satisfactory fit is assured throughout a more or less broad region near both ends of the range of variables. The fit will then be good everywhere Sec. 4-11] PRELIMINARY FIT TO A MONOTONIC FUNCTION 83 if the given function is well adapted to mechanization by an ideal har- monic transformer with the chosen value of AXi. If the chosen value of AXi is not appropriate, the central portion of the generated function, having been subject to no control during this simplified fitting process, may show marked differences from the given function. As an indication of the over-all accuracy of fit attained in this process, and of the appro- priateness of the chosen value of AXi} it is natural to take the difference between the generated and the given functions at the midpoint of the curve, Hi = i; AXi should then be so chosen as to minimize this difference. The following steps can thus be used in obtaining a preliminary fit to a monotonic function: 1. Choose a value of AXif arbitrarily if there is no guide. 2. Choose Xzm and X^m (by a method to be described below) such that the slope of the generated function has the proper values for Hi = 0 and Hi = 1. 3. With these values of the parameters, find the value of H2 when Hi = i. (03 can be read from Table A-2, since AX3 and X3m are known; using this value of 03 to enter the column of Table A-l that corresponds to the known values of AX4 and Xim, interpolate to find the required value of H2.) 4. The difference d between this and the desired value of H2 is taken as a measure of the over-all error in the fit. 5. Repeat the preceding steps for several other values of AXt-, until the trend of d as a function of AXi is established. 6. Choose as the value of AXi to be used in further calculations the one which minimizes \d\. It remains to describe a quick and easy method for finding those values of X3w and Xim for which the generated function has specified terminal slopes: We note that dH2 dH2 _ dlh d<h dds_ dHi dBz ' dHi dlh ( } d$3 For mechanical reasons the input transformer must be such that 03 = 04 = 0 84 HARMONIC TRANSFORMER LINKAGES [Sec. 4-11 0.4 0.2 J -0.2 S •0.4 -0.6 •0.8 y / J ^ . / / '&/ ^l^ /v Y /§/ / // /f, -^^< r y 'p/ ^^ ^^ J^^ S\s^"^, ^\0^ 1Z>^ <?&^ 52= J^— -i -50 ^_- Xi «— 60 .^i0 X; X- -. , "/O -^££.105 -~ ' — — ^Iqq -^**>* go — —- -C«5_ "- -i? ~~ ~_ 40 50 60 70 90 100 110 120 130 140 Fig. 4-17. — Logarithm of the initial slope, (lP)eJ °f ideal-harmonic-transformer functions, plotted against angular travel for a series of values of Xim. Dashed lines indi- cate that the initial slope is negative. Sec. 4-11] PRELIMINARY FIT TO A MONOTONIC FUNCTION 85 ideal-harmonic-transformer —777 ) 1 of i< ad /e=.i functions, plotted against angular travel for a series of values of Xim. Dashed lines indi- cate that the final slope is negative. 86 HARMONIC TRANSFORMER LINKAGES [Sec. 4- 11 when Hi = 0, and 03 = 04 = 1 when H\ = 1. Thus 03=0 and 1 \dHjHl=1 (dHs\ \dd3Je3=i (dHi\ (61) In other words, each terminal slope of the graph of the double-harmonic- transformer operator (#2|#i) is equal to the corresponding terminal slope for the output operator (^2^3) divided by that for the input operator (#i|03). Our problem is thus, in effect, to pick out of the part of Table A-l that corresponds to a given value of AX; two columns such that the ratio of their initial slopes is £0 and the ratio of their final slopes is Si. Consider now Fig. 4-17, which shows the variation with AXt of the quantity logl0 £g) = log10 ((. Y™sXir- Y, )> (62) \d0/9~o \(sin Xi)mai - (sm Xi)min/ for a series of values of Xim. On this chart the distance along the vertical line AX = AX; from the curve X;m = X3m to the curve X;m = X4m (counted as positive upward, negative downward) is ,oglo5o = loglo(g)^_loglo(^o, (63) the logarithm of the initial slope ( -jj^ ) for an ideal double harmonic \atL i/hi=o transformer characterized by the parameters AX,-, X3m, X4m. Con- versely, if we draw a line of length logio $0 on a strip of paper and move this, always in a vertical position, over Fig. 4-17, its ends will continually indicate the paramaters AX,, X3m, and X4m for an ideal double harmonic transformer with initial slope ( -j^- ) equal to the chosen value of £0. \aHi/ H 1=0 Fig. 4-18 presents in a similar manner values of l0gl° (§)-i = l0gl° [(smX)l0-X(dnXu]- (64) It is obvious that if we draw a line of length logio Si on a strip of paper and move it, always in a vertical position, over Fig. 4-18, its ends will Sec. 411] PRELIMINARY FIT TO A MONOTONIC FUNCTION 87 continually indicate the parameters AX», X3m, and X4m for an ideal double harmonic transformer with terminal slope ( %-J ) equal to the chosen value of Si. In order to determine the parameters of an ideal double harmonic transformer for which the initial and terminal slopes have values S0 and Si respectively, one may proceed as follows. (Attention will be restricted to cases in which So and Si are both positive; a case in which both slopes are negative can be reduced to this case by replacing X4m by X4m + 180°.) At the edge of a strip of paper draw an arrow of length |logi0 Sq\ (using the scale at the left of Fig. 4-17) and place it on Fig. 4-17, directing it upward if logio So is positive and downward if this is negative. Similarly construct an arrow of length |logi0 Si\ and place it on Fig. 4-18, directing it upward or downward according as logio Si is positive or negative. If these arrows are placed on vertical lines corresponding to the same AX = AX;, with the heads of both arrows on curves corresponding to the same Xm = X4m and the tails on curves corresponding to the same Xm = Xzm, then these values of AX*, X3m, and X4m give simultaneously the desired initial and final slopes. Such positions for the arrows can be found quickly, for any specified AX;, by placing the tails of the arrows successively at several values of X3m, until a value is found for which the heads of the arrows also lie at the same X4m. Example: As our principal example of double-harmonic-transformer design we shall take the problem of mechanizing the relation x2 = tan xi, (65) previously considered, over the larger range 0° ^ xi ^ 70°, 0 ^ x2 ^ 2.7475. On introduction of homogeneous variables hi = ^o>, h2 - ^ft (66) this relation becomes 2.7475/i2 = tan (hi • 70°) (67) This is tabulated for uniformly spaced values of h i in Table 4-4. The slope of the curve in homogeneous variables is ^ = 0.4447 sec2 xh (68) and the terminal slopes are 0.445 and 3.802. For a preliminary fit we try AX* = 90°. We place on the correspond- ing line in Fig. 4-17 an arrow of length |logi0 0.445|, and on that line in 88 HARMONIC TRANSFORMER LINKAGES [Sec. 411 Table 4-4. — x2 = tan xh 0 = xx ^ 70°, in Homogeneous Variables 0 = xx ^ 70°, in h, h2 0.0 0.0000 0.1 0.0447 0.2 0.0907 0.3 0 . 1397 0.4 0.1935 0.5 0.2549 0.6 0.3277 0.7 0.4187 0.8 0.5396 0.9 0.7143 1.0 1.0000 Fig. 4-18 an arrow of length logio 3.802. We note that if X3m = 10°, cor- rect initial slope requires Xim = —58° (Fig. 4-17), and correct final slope requires X4m = —52° (Fig. 4-18); if X3m = —15°, correct initial slope requires X4.m = —62°, correct final slope requires X4m — —75°. Inter- polating to zero difference of the values of X4m, we have a set of constants assuring correct terminal slopes: AXi = 90°, X3m = -12°, X4m = -59°. Assuming these constants, we now compute H2 for Hi = ?. First we center attention on the input harmonic transformer and determine 03 = 04 : in Table A-2, AXi = 90°, we interpolate between columns for Xim = -15° and Xim = -10°; for H = i, Xim = -12° we find Oi = 0.385 = 03 = 04. Turning attention to the second transformer, we can now determine H2: interpolating between columns of Table A- 1 for AXf = 90°, Xim = — 60° and Xim = -55°, we find that H = 0.325 when Xim = -59° and Bi = 0.385. The desired value of tf2, read from Table 4-4, is 0.255; the curve thus fitted lies too high in the center by d = 0.070. Next we try AX; = 70°. Moving the arrows to the corresponding lines of Figs. 4-17 and 4-18, we find that correct terminal slopes are obtained by using AXi = 70°, X3m = 8°, X4m = - 56°. With these constants, if #i = i, then 03 = 0.371, H2 = 0.308, d = 0.053. Trial of still smaller values of AXt shows that d can be decreased only slightly below this value ; an exact fit of terminal slopes will always lead to a generated curve too high in the middle. The "best" value of AX», in this sense, is a little smaller than is mechanically desirable, and not much can be gained by adopting precisely this value instead of a larger and more convenient one. In the further discussion of this problem we shall therefore fix AX< = 90°. Sec. 412] PRELIMINARY FIT TO A NONMONOTONIC FUNCTION 89 4-12. Preliminary Fit to a Nonmonotonic Function. — Nonmonotonic functions that can be generated by an ideal double harmonic transformer possess only a single maximum or minimum. Expressed in homogeneous variables, they fall into four types illustrated in Fig. 4-19: (a) H2 = 0 (b) H2 = 0 (c) H2 = 1 (d) H2 = 1 when Hi = 0. when Hi = 1. when Hi = 0. when Hi = 1. As with monotonic functions, it is possible to find, for any given AXif values of X3m and X4wl that make the terminal slopes of the generated function equal to those of a given nonmonotonic function. However, a fit of the value of the generated function to that of the given function is assured at only one end of the range of Hi: for Hi = 0 with types (a) and Fig. 4-19. — Types of functions mechanizable by an ideal double harmonic transformer. (c), and for Hi = 1 with types (6) and (d). Agreement of the slopes at the other end of the range of Hi thus does not assure tangency of the given and the generated functions, and the fit may be very unsatisfactory- For this reason it is not advisable to make a preliminary fit to a given nonmonotonic function by the method of Sec. 441. It is usually best to choose a value of X4m such that a fit in the value of the function is secured at the end where this is not otherwise assured, and then make the maxi- mum or minimum in H2 occur for the proper value of H\\ as an indication of the accuracy of the over-all fit one can take the difference between the given and generated functions at a chosen point between the maximum or minimum and the more remote end of the range of Hi. The procedure for securing a preliminary fit to a nonmonotonic func- tion is then as follows: 1. Choose a value of AX», arbitrarily if necessary. 90 HARMONIC TRANSFORMER LINKAGES [Sec. 412 2. Referring to Table A-l, choose X4m such that H2 has the desired value when 03 = Hi = 1, for types (a) or (c), or when 03 = Hi = 0, for types (b) or (d). 3. From the same column of Table A-l read the value of 03 for which Hz = 1 [types (a) or (6)] or the value of 03 for which H2 = 0 [types (c) or (<*)]. 4. From the given function, determine the value of Hi for which Hz = 1 [types (a) or (6)] or the value of Hi for which H2 = 0 [types (c) or (<*)]• 5. By reference to Table A-l or A-2, for the same AX;, find the value of Xzm for which the value of 03 determined in Step (3) corresponds to the value of Hi determined in Step (4). 6. For these values of AX;, X3m, and X4m, determine the difference d between the generated function and the given function at the chosen test* value of Hi. 7. Repeat the preceding steps for several other values of AX;, until the trend of d as a function of AX; is established. 8. Choose as the value of AX; for use in further calculations that which minimizes \d\. Example: As an example, we take the problem of making a preliminary fit to the curve (H*\Hi) of Fig. 4-16 — a case in which we happen to know that an exact fit can be obtained. The curve is of a borderline type, belonging to types (a) and (6) . For the purposes of the preliminary fitting we desire Hz = 0 when Hi = 0, Hz = 0 when Hi = 1, H2 = 1 when Hi = 0.38. For test purposes, we shall compare the generated function with the given function when Hi = 0.70 (desired value, H2 = 0.710). First, choose AX; = 70°. In Table A-l we find that H2 = 0 for both $3 = 0 and 03 = 1 if X4m = 55°; from the same column we see that Hz = 1 for 03 = 0.5. The desired X3wi must then make 03 = 0.5 cor- respond to Hi = 0.38. From Table A-l it is evident that -75° < X3m < -70°; interpolating, we obtain X3m = — 72°. To test the over-all fit given by AX; = 0.70, X3m = -72°,X4w» = 55°, we compute H2 for Hi = 0.70. Interpolating in Table A-2 (since Hi has a value appearing there) between columns corresponding to X3m = —70° and X3m = -75°, we find 03 = 0.771. Returning to Table A-l, X4m = 55 , Sec. 413] A METHOD OF SUCCESSIVE APPROXIMATIONS 91 we obtain by linear interpolation H2 = 0.692 for 03 = 0.771. Linear interpolation, however, is here obviously inadequate ; quadratic interpola- tion yields H2 = 0.700, d « -0.010. Repeating the process with AX; = 90° we find that little interpolation is necessary. To make H2 = 0 for 03 = 0 and for 03 = 1 requires X4m = 45°; the maximum comes for 03 = 0.5, H\ = 0.38; hence Xzm = —75°. Computing H2 for Hi = 0.7, we obtain essentially the graphically deter- mined value, 0.710, and d ~ 0. Although AXi = 90° is the best value, it is evident that the fit is not very sensitive to the choice of AX;. 4»13. Improvement of the Fit by a Method of Successive Approxima- tions.— A satisfactory fit of the generated to the given function is not assured by the simple and rather arbitrary methods just described; these should be depended upon only in choosing a value of AXt-. The final adjustment of X3m and X4m, to obtain the best over-all fit possible with the chosen AXt-, is most satisfactorily accomplished by a graphical method of successive approximations which gives a complete view of the fit at each stage of the process. Convergence of the successive approximations on the final result can be speeded up by exercise of the superior judgment of an experienced designer, but a satisfactory result is assured even for a beginner. The problem to be solved is that of finding ideal harmonic trans- former operators (H2\03) and (03|#i), both corresponding to the chosen AXi, which make the approximate relation (tf2|03) • (03|#i) = (#2|#i) « (/*2|/>i) (69) as nearly exact as possible over the entire range of variables. This will be done by alternately improving the choice of the two harmonic-trans- former operators — that is, the choice of the parameters X4m and X3m, respectively. Let the harmonic-transformer operators chosen after S stages in the approximation be (H2\6S)S and (dz\Hi)s. Then (H2\d3)s • (03|#i)* ~ (hzlhi). (70) Let it be desired to replace (#2|03),s by an operator (#2|03)s+i, which will make the approximation of Eq. (70) more exact. Let the operator Zs be defined by (htlhi) • (tf i|08)s = Z8. (71) Then Zs « (Hi\09)s, (72) 92 HARMONIC TRANSFORMER LINKAGES [Sec. 413 as may be shown by multiplying Eq. (70) from the right by the operator (Hi\d3)s. If this approximation were exact, Eq. (70) would necessarily be exact; if this approximation is improved, that of Eq. (70) will be improved. Now Zs can be computed with sufficient accuracy by graphi- cal methods. If it is possible to find an ideal-harmonic-transformer operator (#2|03)s+i which gives a better fit to Zs than does (#2|03)s, then this is the desired improved operator; the approximation in the relation (#.|0a)a+i-(0s|ffi)s « (*i|*i) (73) is better than that in Eq. (70). Next one will wish to replace (03|i7i)s by an improved operator (03|#i)s+i. Let the operator Ys+i be defined by (h2\hi) • Ys+i = (H2\d3)s+1. (74) By Eq. (73) (Hi|0s)a « Ys+i. (75) An improved operator (Hi\$a)s+i would make this approximation more exact; one can therefore determine it by computing Ys+i by graphical means and finding the ideal harmonic transformer function that best fits this function. The approximation in writing (H2\d3)s+1 • (03|# iWi « (Mfci) (76) is then even better than that in Eq. (73). It is now possible to make a further improvement in (i72|03), comput- ing Zs+i by Eq. (71) and fitting (#2|03)s+2 to this as exactly as possible. The operator (0t\Hi) can then be improved again, and the process repeated until the improvement obtained does not repay the effort expended. It is of course possible that a satisfactory fit can not be given by any ideal double harmonic transformer; it will then be necessary to make use of methods to be described later in this chapter. Example: We return to the Example of Sec. 4*11, the mechanization of the tangent function from 0° to 70°. We there fixed on the value AXt = 90° and found approximate values of X3m and X4m. Rounding off these values to those appearing in Table A-l, we might take (H2\d3)~X4m= -60° (77) (0s|ffi)~X8)n= "5°. (78) These values, especially the second, are good. In order to provide a better illustration of the method of successive approximations we shall deliberately select a poorer value, X3m = — 15°, with which to start the computations. Figure 4-20 shows a graph of the given function (/i2|/ii), points on the graph of (Hi|08)i ~ -15° ^ X3 ^ 75°, as read from Table A-l, and Sec. 413] A METHOD OF SUCCESSIVE APPROXIMATIONS 93 the construction needed to determine corresponding points on the graph of Zh which has been drawn in as a continuous curve. A good over-all fit to Zi can not be found in Table A-l (AX< = 90°!), but a reasonable fit at the lower end is obtained by taking Xim = —70°, as shown in the same figure. Therefore, as the basis for the next step in the com- putation we make (#2! 0a) 2 correspond to X4m = —70°. < ) y c K ; ■ > y / < / < \ / L / ( _/ i > A / < >/ / (#11*3 )i(-15° -75°) / { / / > , / A / A V < <*' V /\ zx V S < / ^ \r. X „ * \ y 0 (H2\d3) (h2\h 1) 20°) > 6 ^ Fig. 4-20. — Mechanization of xi = tan xi. First step in the method of successive approximations: construction of the operator Zi and approximate fitting of this by (tf2|03)i~AXi = 90°, Xim = -70°. Next, Fig. 4-21 shows the construction used in determining Y2. (In practice this would be carried out on the same graph as the construction for Zi, but for the sake of clarity a new figure is used here.) The operator (#i|03)2 can be made to fit this fairly well by taking X3m = —5°. Repetition of this process will lead to little further improvement. It can be seen in Fig. 4-23 that Z2 is perhaps best fitted by the value of Xtm -70°, arrived at in Fig. 4-20. It would be of little value to reduce the error further by interpolation in the tables; the solution would in any case apply only to an ideal double harmonic transformer, which could be realized only by using undesirably complex mechanisms or 94 HARMONIC TRANSFORMER LINKAGES [Sec. 4- 13 Fig. 4-21. — Mechanization of xi = tan xi. Second step in method of successive approximations: construction of the operator Y2 and approximate fitting of this by (Hi\0i)t~AXi = 90°, Xim = -5°. *4Ar=20° X*=-5° Xi H, 0.0 e3~\ ' I 0.0 I 0.5 1.0 l^-~ H2 I T 1 . ■ 1 1 1 1 1 1 0.0 0.5 1.0 Fig. 4-22. — Ideal double harmonic transformer approximately mechanizing xt ■= tanxi. Sec. 4-14] NONIDEAL DOUBLE HARMONIC TRANSFORMERS 95 nonideal double harmonic transformers with very long links. It is better to design the transformer as nonideal, further reducing the error by adjustment of link lengths and slide positions, as explained in Sec. 4-13. Figure 4-22 shows the ideal double harmonic transformer correspond- ing to the present stage of solution of the problem. The cell has been normalized by making i2i[(sin Xi)max - (sin Xi)min] = #2[(sin X2)max - (sin Z2)mm]. (79) 4*14. Nonideal Double Harmonic Transformers. — The field of mechanizable functions is very substantially extended if nonideal har- monic transformers are coupled instead of ideal ones. (A typical non- ideal double harmonic transformer is shown in Fig. 4-26.) Instead of three independent parameters, there are seven to be adjusted: AX 3 = AX4, X3m, Ximj Lh L2, E*. and E*. Here, as before, the lengths L\ and L2 of the links are to be measured in terms of the horizontal travels AXi and AX2, respectively. E? is the reading on the iJf-scale where it is intercepted by the center line of the X[ slide, and E* is the reading on the H*-sca\e where this is intercepted by the center line of the X2 slide. The Peaucellier inversor shown in Fig. 2-4 is a special case of the non- ideal double harmonic transformer, with X3w = X4m, Li = I/2, and Ef = E% = 0; it is thus evident that such devices can serve for the mechanization of functions that are not even roughly of sinusoidal form. To determine the function generated by a given nonideal double harmonic transformer one can apply the method described in Sec. 4-8, obtaining the operator (#2|fl\) as the product of operators (H2\6z) and (03]#i), which describe the component nonideal harmonic transformers. In the converse problem of mechanizing a given function by a non- ideal double harmonic transformer it is not feasible to vary all seven of the available parameters simultaneously. One should begin as though the double harmonic transformer were to be ideal, carrying out an approximate fit (Sees. 4-11 and 4-12) to determine a value of AXt-, which is held constant thereafter, and then improving the choices of X3m and X4« (Sec. 4-13) until this ceases to be profitable. At this point it becomes necessary to begin the adjustment of Lh L2, Ef , Ef. Since the device may be regarded as two nonideal harmonic transformers in series, the problem to be solved is still formally the same as that considered in Sec. 4-13 — that of making the approximation in (ffi|0») ' (0,|J?i) = (#2|# 1) « (fo|fci) (69) as nearly exact as possible — and the method of solution by successive approximations is the same. Here, however, each of the component transformers is characterized not by one, but by three constants, (X3m, 96 HARMONIC TRANSFORMER LINKAGES [Sec. 4-14 E*, Li) or (Xim, E*, La), which must be chosen at each stage of the process — for instance, by the methods of Sees. 4-4 and 4-8. Example: We continue the example of Sec. 4-13, that of mechanizing the tangent function from 0° to 70°. Figure 4-23 shows the construction of Zz, to fit which we shall now adjust the three constants characterizing (#2 1 #3) : X4mf E*, and L2. As already noted, the best fit obtainable with ( / y\ y < ^1 \ 1 < \ j /j 1 c (#1 ie3)2(-5 °-^85°) < *2 / » \ / \ i f? * < > -p' ^{n2 ^3)3 ("7 )°— 20°; £*=0.5, L — 2) c , / ^^M tf2ie3)2 (-70°-^ 20°) / ^/i i ^^ > 4 Fig. 4-23. — Mechanization of xi = tan x\. Third step in method of successive approxi- mations: construction of the operator Zz and approximate fitting of this by (tf2|03)2~AXt- = 90°, Xim = -70°, (...). and by (ffi|0«)« ~ AX»- = 90°, Xim = -70°, E* = 0.5, L = -2 (crosses). an ideal-harmonic-transformer operator is given by X4w = —70°, X\m = 20°; the residual error then changes sign twice, tending to be large near the ends of the range of variables. Now the limits of X4 here are roughly the same as those of the example of Sec. 4-8 ( — 15°, 75°) except for a change in sign, and the geometrical situations differ only as mirror images if one replaces a link to the left by a link to the right, and vice versa. Correspondingly, one easily sees, the structural error func- tions 8Hk applicable here differ from those of Fig. 4-10 in replacement of Sec. 414] NONIDEAL DOUBLE HARMONIC TRANSFORMERS 97 6i by 1 — 6i, that is, reflection of the curves in a vertical line. Thus it is evident that it is not possible, by any choice of Ef and L2, to obtain a structural-error function that changes sign twice, such as is needed to give a good fit to Z2 over the whole range of variables. We shall there- fore concentrate our attention on improving the fit for low values of 0, raising the curve in this region, and attempting only to keep the change small elsewhere. Inspection of Fig. 4-10 then shows (the differences of Fig. 4-24. — Mechanization of X2 = tan xi. Fourth step in the method of successive approximations: construction of the operator Ys, and approximate fitting of this by an ideal-harmonic-transformer function with AXt- = 90°, X»m = —7.5° (circles) and by (#i|03)3~AXi = 90°, Xim = -7.5°, E* = 0.2, L = 2 (crosses). IS the present from the former case being borne in mind) that E* = an appropriate value, and that L should be negative, the link to the right. Rough consideration of the magnitudes involved leads to choice of L = — 2. The resulting fit, as shown in Fig. 4-23, is quite satisfactory for low values of 0. The process of successive approximations is continued in Fig. 4-24 with the graphical construction of F3. We have now to fit (Hi |03)3 to this by choosing X3m, Ef, L\. Inspection of Table A-l shows that with 98 HARMONIC TRANSFORMER LINKAGES [Sec. 4-14 X3m = — 5° one has a bad fit at the left, and with X3w = — 10° the curve is much too low in the middle. In such a case it is desirable to inter- polate. We choose Xzm = —7.5°. The values of the corresponding function are found with sufficient accuracy for our graphical method by a linear interpolation in Table Al; the resulting values are plotted in Fig. 4-24 as a series of points in small circles. In further improving the fit one will wish to raise the central part of the curve, to depress the 1 c ) . y 7 < 4 H I / / < / 1 / / ^ (tfjlftj 3 (-7.5°- -82.5°; E*=0.2,Z ( = +2) / V — i i 4 < / / n i 5°— -15? =-0.8) / i \ y K, X ( / y < / ^S __\^ A 9 (-75°-^15°) Fig. 4-25. — Mechanization of X2 = tan xi. Fifth step in the method of successive approximations: construction of the operator Z%, and approximate fitting of this by {Hi\B£)i: AXi = 90°, Xim = -75°, E* = 0.42, L = -0.8. extreme upper end, and to leave the lower and unchanged. The values of Xzm and X3M are sufficiently like those of Fig. 4-10 for this to be used as a guide; it is again evident that no choice of constants can accomplish everything that is desired. We choose therefore to allow a considerable error at the lower end of the curve, leaving this to be corrected (as before) by our choice of E* and L2; we concentrate on improving the fit at the upper end, and, secondarily, that in the central region. Inspection of Fig. 4-10 leads to choice of Ef = 0.2 and L = +2. Computation of the structural-error function then leads to the corrected points of Fig. 4-24, Sec. 414] NONIDEAL DOUBLE HARMONIC TRANSFORMERS 99 indicated by crosses. It is evident that a value of Ef nearer to zero would have been preferable, as giving a depression of the curve over a less extended region. However, it is hardly worth while at this stage of the computation to make a more careful choice of constants, and we accept the resulting function as (#i|03)3. The next stage of the calculation, the determination of (^2^3)4, is shown in Fig. 4-25. When Z3 is constructed it is found that a better fit can be obtained at the upper end by taking XZm = — 75° than by taking XZm = -70°. The error functions shown in Fig. 4-10 then apply exactly, with the substitution 0< — > 1 — 0*. Since preliminary fits have been made in all parts of the range of 0,-, it is now worth while to make a careful adjustment of the constants E* and L2, as by the methods of Sec. 4-8. With E% = 0.42, L2 = -0.8, one finds exactly computed points that give an excellent fit except at the extreme upper end of the curve. This is very satisfactory, as it is in this last region that the fit is being controlled by choice of Ef and L\. The final graphical stage of the solution, the determination of (i?i|03)4, is not illustrated by a figure. It leads to the choice of X3« = —7.5°, Ef = 0.2, Li = +1.8, with an excellent over-all fit. This is as far as the fit can be carried by these graphical methods; further refinements are best obtained by the methods discussed in Chap. 7. We have thus arrived at the following choice of constants: Xzm = —7.5°, XZM = 82.5°, Et = 0.2, Lx = 1.8, (80a) X^m = — 75 , X±m = 15 , Ef = 0.42, L2 = -0.8. (806) Calculation of the resultant total structural error is illustrated in Table 4-5, which consists of three sections. The first shows the calcula- tion, by the method described in Sec. 4-6, of values of the homogeneous input parameter H[ for a series of values of 03. The second shows the calculation of values of the homogeneous output parameter Hr2 for the same series of values of 03. In the third section there are shown cor- responding values of xi = H[ • 70°, (81) the generated tangent function x2g = #'2 tan 70°, (82) the ideal tangent function x2 = tan xh and the ideal generated homo- geneous variable h2. The error in the generated function, 8h2 = Hf2 — /i2, is found to be less than 0.8 per cent of the total variation of the output variable. The linkage corresponding to these constants is drawn in Fig. 4-26. Reduction of the linkage to the normalized form shown here requires a 100 HARMONIC TRANSFORMER LINKAGES [Sec. 4- 14 Table 4-5. — Computation of the Total Structural Error (Linkage of Fig. 4-26) H* -Et sin «i DHX #i 03 tfi (H* - E?) 1 — COS ei = LrX (1 — COS Ci) + DHy - (DH1)0 #i (Ht\**)4 0.0 0.0000 0.7887 0.3396 0.0594 0.1069 0.0000 0.0000 0.1 0.1398 0.7986 0.3438 0.0610 0.1098 0.1427 0.1586 0.2 0.2791 0.7801 0.3358 0.0581 0.1046 0.2768 0.3076 0.3 0.4143 0.7337 0.3159 0.0512 0.00 2 0.3996 0.4441 0.4 0.5421 0.6605 0.2844 0.0413 0.0743 0.5095 0.5662 0.5 0.6586 0.5624 0.2421 0.0297 0.0535 0.6052 0.6726 0.6 0.7633 0.4418 0.1902 0.0183 0.0329 0.6893 0.7661 gi =0.7749 0.7 0.8513 0.3016 0.1298 0.0085 0.0153 0.7597 0.8443 0.8 0.9211 0.1453 0.0626 0.0020 0.0036 0.8178 0.9089 ^=0.4305 Li 0.9 1.0 0.9711 1.0000 -0.0233 -0.2000 0.0100 0.0861 0.0000 0.0037 0.0000 0.0067 0.8642 0.8998 0.9605 1.0000 03 H2 H$ -Ef sin c2 1 — COS C2 DH2 H2 + DH2 Hi - (DH2)0 (#2|03)4 0.0 0.0000 -0.4200 -0.3178 0.0518 0.0414 0.0000 0.0000 0.1 0.0428 -0.2204 -0.1667 0.0140 0.0112 0.0730 0.0750 0.2 0.1038 -0.0344 -0.0260 0.0003 0.0002 0.1450 0.1490 0.3 0.1819 0.1336 0.1011 0.0051 0.0041 0.2192 0 . 2252 g2 = 0.6052 0.4 0.2748 0.2793 0.2113 0.0226 0.0181 0.2981 0 . 3063 0.5 0.3804 0.3992 0.3020 0.0467 0.0374 0.3844 0.3950 0.6 0.4961 0.4904 0.3710 0.0714 0.0571 0.4804 0.4936 ~ = 0.7564 Li 0.7 0.6189 0.5505 0.4164 0.0908 0.0726 0.5877 0 . 6039 0.8 0.7459 0.5782 0.4374 0.1007 0.0806 0.7067 0.7262 0.9 0.8740 0.5726 0.4331 0.0986 0.0789 0.8365 0 . 8595 1.0 1.0000 0.5340 0.4040 0.0852 0.0682 0.9732 1.0000 03 X\, degrees X2g tan X\ ha Shi 0.0 0.00 0.0000 0.0000 0.0000 0.0000 0.1 11.10 0.2061 0.1962 0.0714 0.0036 0.2 21.53 0.4094 0.3945 0.1436 0.0054 0.3 31.09 0.6187 0.6030 0.2195 0.0057 0.4 39.63 0.8416 0.8282 0.3014 0.0049 0.5 47.08 1.0853 1.0754 0.3914 0.0036 0.6 53.63 1.3562 1.3579 0.4942 -0.0006 0.7 59.10 1.6592 1.6709 0.6081 -0.0042 0.8 63.62 1.9952 2.0163 0.7339 -0.0077 0.9 67.24 2.3615 2.3836 0 . 8675 -0.0080 1.0 70.00 2.7475 2.7475 1.0000 0.0000 Sec. 415] ALTERNATIVE METHOD OF TRANSFORMER DESIGN 101 slightly different calculation from that in the case of ideal double har- monic transformers (Eq. 79). Perhaps the simplest method is to choose arbitrary values of Ri and R2, and to compute the corresponding travels AX[ and AX2 from the geometry of the linkage. Since these travels are proportional to the R's, and are to be equal in the normalized cell, one has as the ratio of the normalized arm lengths Rln R2n gi AX2 R2 AX[ (83) AX, AX Fig. 4-26. — Nonideal double harmonic transformer generating, approximately, X2 = tan xi, 0 < xx < 70°. Values of the constants are given in Eq. (80). 4«15. Alternative Method for Double -harmonic -transformer Design The graphical method described in Sec. 4-14 has the advantage that it permits readjustment of the constants X3m and Xim at all stages of the design process. The alternative method to be described in the present section is useful when values of AX3, X3m, and Xim can be considered as fixed; it is essentially an extension of the method of Sec. 4-8, which per- mits simultaneous adjustment of the constants Lit L2) E*t E*, of the two harmonic transformers. Let us assume that a given relation h2 = (^2|^i) ' hi (84) has been mechanized approximately by an ideal double harmonic trans- former that generates the relation Ht = (#2|#i) • H1 (85) 102 HARMONIC TRANSFORMER LINKAGES [Sec. 415 between its input and output parameters. This relation we can express parametrically in terms of the angle 03: Hi - #1(03), Hi = H%(dz). (86) Without changing the constants AX3, X3m, -X"4m of this linkage, let the ideal harmonic transformers be replaced by nonideal ones. The input and output parameters will then be H[ and H'2, differing from Hi and Hi by the structural-error functions 5Hi and 5H2: HW*) = #l(03) + «&!(*), } #$(03) = #2(03) + 5tf2(03). J K*n The resulting nonideal double harmonic transformer will then generate a relation Hf2 = (H'2\H[) • H[. (88) Our problem is to assign to the constants Lh L2, Ef, Ef, values such that Eq. (88) will approximate as closely as possible to the given relation, Eq. (84), when H[ takes over the role of Hh H2 that of H2. It was shown in Sec. 4-8 that when 8Hi and 5#2 are small one can write 8H1 = a/i(03) + 6/2(03), (89a) 8H2 = c/3(03) + d/4(03), (896) where (90) The functions /i(03) and/2(03) can be computed using Eqs. (39) and (40), with Hk replaced by H 1(03)5/3 (03) and/4(03) are also computed by Eqs. (39) and (40), respectively, with Hk replaced by #2(03). Let it be desired to choose the constants a, 6, c, d, in such a way that the linkage generates the desired relation exactly for some fixed value of 03 : /ii(03) - # i(03) + a/i(03) + 6/2(03), (91a) A2(03) = #2(03) + c/3(03) + ay4(03). (916) Equation (84) specifies which values of /ii and /i2 must correspond to each other, but there is nothing to prescribe which pair of values (hi, A2) must correspond to any given value of 03. We could, for instance, pick this pair arbitrarily and still satisfy Eqs. (91) by properly choosing the disposable constants. However, we do know that if Eqs. (91) are to be accurate dHi and 8H2 must be small; /ii(03) must be nearly equal to -A 2£, ° 2U a " 2L2 Sec. 415] ALTERNATIVE METHOD OF TRANSFORMER DESIGN 103 Hi(ds), hi{6z) nearly equal to Hz{Bz). We therefore place on our choice of the pair of values (hh h2) only the condition that hi(B9) = Hi(Ot) + Ahh (92) where Ahi is small. The corresponding value of ^2(^3) is easily computed. Let h2 = hf(6z) when hi = #i(03). (93) Then \dhi/ hi=Hi(d3) h(e3) = ^»)(93) + l-f -Ahh (94) \arli/ hi=Hi($3) to terms of the first order in the small quantity Mi. Combining Eqs. (91,) (92), and (94), we find that the- conditions to be satisfied are a/i(03) + bMOt) = Ahh (95a) c/3(03) + dMdt) = fcf(08) - H2(B3) + ^ • Ahi. (956) Eliminating Ahi from these equations, we have = fcg»(0.) - Ht(0>). (96) This is the only condition that must be satisfied by the constants a, b, c, d, so long as no attempt is made to control the value of the small quantity Ahi. Since one can satisfy simultaneously four conditions such as Eq. (96), it is possible to make the linkage generate the desired relation exactly at four chosen values of #3. One has to solve four simultaneous linear equations in the four unknowns o, b, c, d: = h?>(6«>) - H,(«p), (97) The constants of the linkage can then be computed by Eqs. (90). This should be followed by exact calculation of the function generated by the linkage, as in the example of Sec. 4- 14. Example. — To illustrate this method we shall treat again the example considered in Sec. 4-14. Here, however, we shall accept as fixed the 104 HARMONIC TRANSFORMER LINKAGES [Sec. 4-14 constants arrived at in Sec. 4-13, AX3 = 90° X 3m — — O Xim = -70°, (98) fcf. Fig. 4-22) and shall adjust only the constants Lh L2, Ef, E%. The function #2(#i) generated by the linkage of Fig. 4-22 can be written down in parametric form by reference to Table A-l, the values of Hi being found in the column AX; = 90°, Xim = — 5°; the values of H2, in the column AX; = 90°, Xim = —70°. These are shown in Table 4-6, together with the corresponding values of /i20) [computed by Eqs. (65) and (66) with hi set equal to Hi] and the over-all structural error. The structural-error function exceeds 3 per cent of the total travel; by choice of the four disposable constants we shall now attempt to reduce this error to zero for Of = 0.2, 0.4, 0.6, and 0.8. Table 4-6. — Function Generated by Linkage of Fig. 4-22 03 Hi H2 h™ &<!' - Ht 0.0 0.0000 0.0000 0.0000 0.0000 0.1 0.1448 0.0508 0.0650 0.0142 0.2 0.2881 0.1183 0.1337 0.0154 0.3 0.4262 0.2010 0.2087 0.0077 0.4 0.5559 0.2969 0.2938 -0.0031 0.5 0.6738 0.4034 0.3926 -0.0108 0.6 0.7771 0.5181 0.5083 -0.0098 0.7 0.8632 0.6381 0.6413 0.0032 0.8 0.9301 0.7604 0.7843 0.0239 0.9 0.9761 0.8820 0.9159 0.0339 1.0 1.0000 1.0000 1.0000 0.0000 We have first to give explicit numerical form to Eqs. (97), which determine the constants a, b, c, d. Values of Hf and H$ are read from Table A-l, for the chosen values of 03; the /'s are then computed as explained below Eq. (90). Values of -^ can be computed by Eq. (68) Table 4-7. — Constants Required in Design Procedure 03 m /i(03) /2(03) #2 Mes) fM dhz dh\ 0.0 0.9958 0.0000 0.2 0.9719 0.2587 -0 . 5260 0.4159 0.0754 -0.6169 0 . 5046 0.4 0 . 8435 0.2836 -0.8025 0.7402 0.3030 -0.9410 0.7344 0.6 0 . 6232 0.1736 -0.8025 0.9411 0.4582 -0.9410 1.3121 0.8 0 . 3326 0.0433 -0.5260 0.9991 0.3709 -0.6169 2.5094 1.0 0.0000 0.9083 Sec. 415] ALTERNATIVE METHOD OF TRANSFORMER DESIGN 105 with xi = 70° X Hi. All these quantities appear in Table 4-7. By using also the last column of Table 4-6 we can easily determine all the constants of Eqs. (97) : -0.1305a + 0.26546 + 0.0754c - 0.6169d = 0.0154 ■0.2083a + 0.58946 + 0.3030c - 0.9410d = -0.0031 ■0.2278a + 1.05306 + 0.4582c - 0.9410a7 = -0.0098 ■0.1087a + 1.31996 + 0.3709c - 0.6169a1 = 0.0239 (99) Solution of these equations yields a = 0.1966, 6 = 0.0566, c = -0.1874, Hence, by Eqs. (90), Li = 1.806, L2 = -0.703, Ef = 0.288, d = -0.0651. (100) E* = 0.347. (101) The constants specified by Eqs. (98) and (101) are not very different from those found in Sec. 4-14, and the linkage would closely resemble that of Fig. 4-26. The exact total structural error of the new linkage is given in Table 4-8: it is about a third of that of the first design. At first sight it may appear surprising that the error does not vanish for 03 = 0.2, 0.4, 0.6, 0.8, since this was the condition applied in determining the constants of the linkage. It is to be remembered that the equations on which this method is based are approximations obtained by treating e Table 4-8. — Total Structural Error in Second Mechanization of x2 = tan xi, 0 < X! < 70° 03 H[ = hi m h2 5/l2 0.0 0.0000 0.0000 0.0000 0.0000 0.1 0.1589 0.0734 0.0715 0.0019 0.2 0.3074 0 . 1460 0.1435 0.0025 0.3 0.4429 0.2212 0.2187 0.0025 0.4 0.5643 0.3020 0.3001 0.0019 0.5 0.6709 0.3908 0.3899 0.0009 0.6 0.7628 0.4905 0.4901 0.0004 0.7 0.8409 0.6024 0.6026 -0.0002 0.8 0.9061 0.7268 0.7280 -0.0012 0.9 0.9589 0.8609 0.8624 -0.0015 1.0 1.0000 1.0000 1.0000 0.0000 as a small angle. The error computed by these formulas does vanish at the specified values of 03, but there are present other and larger errors due to the use of the small angle approximations, which are, essentially, those disclosed by the exact calculations on which Table 4-8 is based. We could make allowance for these errors, approximately, by repeating the calculation, taking as the constants on the right-hand side of Eqs. 106 HARMONIC TRANSFORMER LINKAGES [Sec. 4-15 (99) sums of corresponding entries in the last columns of Table 4-6 and 4-8. Such a refinement would be worth while only if the mechanism were to be constructed with exceptional care. In Sec. 6-6 we shall meet a problem in which the straightforward application of this method leads to a less satisfactory result; the required modification of the method will be described there. CHAPTER 5 - THE THREE-BAR LINKAGE 5-1. Fundamental Equations for the Three-bar Linkage. — A three-bar linkage (Fig. 5-1) consists of two cranks, Bh A2, pivoted on a frame and connected through a link B2. The symbols Ah Bh A2, B2 will be used to represent distances between the pivotal points within the correspond- ing mechanical parts: Bi and A2 are the lengths of arms of the cranks, B2 is the length of the connecting link, and Ai is the distance between pivots in the frame. The three-bar linkage serves as a mechanical cell having one of the cranks as the input terminal, the other as the output terminal. The Fig. 5*1. — Symbols used in the discussion of three-bar linkages. input and output parameters, Xi, X2, are rotations of those cranks measured clockwise from a zero line passing through the pivotal points, Si and S2, of the cranks; the zero position for each crank is that in which it points toward the left. The functional relationship of the parameters X1} X2 follows from the geometry of the quadrilateral SiTiT2S2. To find X2 graphically for a given Xi and dimensions At, Bh A2, B2) one would first construct the zero line S\S2 and the line of the input crank, S1T1. The end T2 of the output crank must lie on a circle of radius B2 about Tij and on a circle of radius A2 about S2. If these two circles intersect, a solution for X2 exists; in general they will intersect in two points T2+, T2~, which are vertices of two congruent triangles, TiS2T2+ and TiS2T2-. Let rji be the principal value of the angle S&Ti, lying 107 108 THE THREE-BAR LINKAGE [Sec. 5-2 between —180° and ' + 180°, and 772 the principal value of the angle T1S2T2+, lying between 0° and +180°. There are then two possible values of X2: X2+ = 1)1 + V2, X2- = 771 — 772. (1) In terms of the problem specified here, X2 is thus a double-valued func- tion of Xij the functional relation X2 = (X2|Xi) • Xi has two branches, represented by the operators (X2+|Xi) and (X2_|Xi). If 771 is not restricted to its principal value, X2 is of course a highly multiple-valued function of Xi. Cases in which this multiple- valuedness is of significance in actual mechanical cells will appear later. For numerical calculation of X2 the following procedure is probably the best: 1. Compute the diagonal D of the quadrilateral using the cosine law: D2 = A\ + B\ + 2A1Bl cos Ii. (2) 2. Compute 771 and 772 by further applications of this law: £)2 _J_ £2 _ g2 cos 771 = oTT) ~' ^k s*n r}1 s*n Xi > 0, (3) 7)2 I A2 _ D2 cos 772 = 2AD ™th 0 < 772 < 180°. (4) 3. Find X2+, X2_ by Eq. (1). 5*2. Classification of Three-bar Linkages. — Three-bar linkages are conveniently classified according to the inherent limitations on the range of the input parameter Xi. To find these limits, within which the func- tion X2(Xi) is defined, we observe first that the diagonal D is a side of the triangle S1S2T1 and as such is limited: \Ai - Bi\ ^D ^ Ai + £1. (5) Similarly, since D is a side of the triangle 77i*S27T2+, \At - B2\ ^ D ^ A2 + B2. (6) These are the only limitations on D, and they imply the limitations on Xi with which we are concerned. Since Eqs. (5) and (6) apply simultaneously, they must be consistent; unless there is overlapping of the intervals set by them for D, it will not be possible to construct a cell with the given dimensions. When the two intervals overlap, D can take on any value within the range common to them. As is illustrated in Fig. 5-2, the intervals can overlap in four different ways, which form the first basis for our classification of these linkages: Sec. 5-2] CLASSIFICATION OF THREE-BAR LINKAGES 109 Class a: \A2 - B«| < \Ai - A| < Ax + Bt < A2 + B2, (7) Class b: \Ai - Bx\ < |A, - B>| < Ax + J5i < A2 + B2, (8) Class c: \A2 - B2\ < \Ax - #i| < A2 + B2 < At + £i, (9) Class d: \Ai - BJ < |A2 - B2| < A2 + B2 < Ax + Bi. (10) In each case, D can take on all values between the two intermediate quantities of the corresponding line. lA^BJ A1*Bl The linkages of Class a have an class a h~~" ~~*__ unlimited input, since Eq. (5) implies \A2-B2\ Az+B2 no limitation on Xh and Eq. (6) is automatically satisfied. Linkages of \a-bi ^i+-Bi the other three classes have a limited I 1 TTT ..,,., r Class 0 . I mput range. With linkages ol u rj-j ^_I+B Class 6, passage through the value Xi = 180° is impossible, since D can- not assume the corresponding value \AX-B^ Ai^Bi | Ai - Bi\. With linkages of Class c, Class c , __, passage through Xi = 0° is excluded, \A2-B2\ A2+B2 since D cannot assume the cor- responding value A i + B i. Finally, lAj-ffil ^i+^i with linkages of Class d, passages class d through Xi = 0 and Xi = 180° are \A2-B2\ A2+B2 both impossible; D cannot attain Fig. 5-2. — Classification of three-bar values corresponding to either of linkages. these points. The range of the output variables can be discussed similarly. From what has been said it is obvious that the four linkages with 1. Ax = p, Bx == q, A2 = r, B2 = s 2. Ax = q, Bx = p, A2 = r, B2 = s 3. Ax = p, Bx = g, A2 = s, jB2 = r 4. Ai = g, Bx = p, A2 = s, B2 = r belong to the same class. Now the relative magnitudes of Ax, A2, Bx, B2 form the basis of a further subclassification of three-bar linkages, the subclasses being given the numerical designation above if one takes always p > q, r > s. That is, Class a linkages are divided into four subclasses : al:Ax>Bx, A2 > B2, (11) a2: Ax < Bx, A2 > B2, (12) a3: Ax > Bx, A2 < B2, (13) a4: Ax < Bx, A2 < B2, (14) and the other classes are similarly divided into four subclasses. 110 THE THREE-BAR LINKAGE [Sec. 5-2 V \ SX + \ h \ L°n \ 0 ^ Lfc i 1 r* ■"N / i ) o\ ^J x Ite Sec. 5-2] CLASSIFICATION OF THREE-BAR LINKAGES 111 o eg 9 <V) <o >< CM f\ ( V V 1 o 112 THE THREE-BAR LINKAGE >ec. 5-3 Finally, in each subclass X2 is a function with two branches, X2+ and X2-, which we place in separate sub-subclasses of the subclass. Three- bar linkages are thus divided into 4 X 4 X 2 = 32 sub-subclasses in all. A sub-subclass will be indicated by a symbol such as c3+, which applies to the positive branch of a linkage for which 14 B, < \At - Bi| 4i > Bh <A2 + B2< Ai + Bh A2 < B2. The general forms of the functions generated by all these types of three-bar linkage are illustrated in Fig. 5-3. In each case the X2 has been plotted as a function of X\, for a three-bar linkage with dimensions illustrated in the adjoining sketch. A mechanical configuration and the generated curve are both shown for the positive branch by continuous lines, for the negative branch by dotted fines. The value of X2 shown is not necessarily the principal value. In some cases the positive and negative branches join continuously, but always at a point of infinite slope near which the linkage is not operable. The reader should study this figure carefully, since one should not attempt to mechanize by this means functions that obviously are not included in the class of functions of the three-bar linkage. 5*3. Singular Cases of Three -bar Linkages. — Certain special three- bar linkages that belong to more than one of the classes defined above, Fig. 5-4. — Three-bar linkage with A as limiting cases, have special properties that entitle them to separate mention. Case A: Ai + Bi = Ai + Bi (15) A linkage of this type (Fig. 5-4) has a singular point for Xi = 0. So long as the input variable is restricted to a range not including the point Xi = 0, the configuration of the mechanism and the value of the output variable are uniquely determined. When X\ = 0 the value of X2 is still uniquely determined, but the mechanism has at this point an inde- terminate motion, there being two possible finite values for dX2/dX\. Sec. 5-3] SINGULAR CASES OF THREE-BAR LINKAGES 113 Thus, when the input parameter is allowed to pass through the value Xi = 0, X2 may or may not pass from the positive to the negative branch of the function, or conversely; the value of X2 is no longer uniquely determined by the value of Xi, but may have either of two values, unless appropriate stops are introduced. Case B: A1 - Bt\ \Ai B, (16) In this case (Fig. 5-5) a similar singularity exists for X\ = 180( X2 Fig. 5-5. — Three-bar linkage with A\ — B\ = At — B2. Case C: Ax + Bx = A2 + B2 \ • u , /17N In this case there are, of course, singular points for both X\ = 0 and Xi = 180°, as well as some other important features that should be mentioned. The conditions in Eq. (17) can be satisfied in two ways: d: Ai-'Bi = -(A2- B2); Bx = A2; Ax = B2. (18) C2: Ax - B1 = A2 - B2; Ax = A2; Bx = B2. (19) The Case C1} the parallelogram linkage (Fig. 5-6), is very well known. Its positive branch (for 0 < Xi < 180°) is used to transmit rotation from one shaft to another at the ratio 1 to 1, within limits set far enough from the points of singularity, at which backlash may become important. (A good range in practice is 30° < Xx < 150°, but larger ranges can be attained by increased care in manufacture.) The corresponding nega- tive branch of the linkage function, shown dotted in Fig. 5-6, is rarely used; its curvature decreases as the length of the link B2 is increased. It will be noted that the various positive and negative branches, differing by changes in Xi and X2 which are multiples of 2tt, form a connected network through the whole of the XiX2-plane. If no stops are intro- duced the generated X2 may or may not pass from a positive branch to a 114 THE THREE-BAR LINKAGE [Sec. 5-3 negative branch, or vice versa, every time Xi passes through a value that is a multiple of it. The value of X2 is thus not uniquely determined by the value of X%) it is not even restricted to one of two values, as in Cases A and B ; it may take on an infinite number of values, which fall, of course, into two sequences with spacing 2t, corresponding to the positive and negative branches. Fig. 5-6. — Three-bar linkage with Bi — At, Ai = Bi. Linkages of Class Ci (Fig. 5-7) are of special interest in that X2 remains zero on part of the positive and negative branches, whatever the value of X\) how this can happen will be evident from the geometry of the sketch. [The classification of branches as positive and negative is here quite formal ; physically it would be more appropriate to think of the branches as (1) the straight line X2 = 0, and (2) the oscillatory curve with continuous derivative.] If the generated X2 is following the positive !■■ A v ^ * — * Fig. 5-7. — Three-bar linkage with A\ = Ai, B\ = B2. branch between Xi = 0 and Xi = ir, and X\ passes through the former point, then X2 may continue to change at a uniform rate by passing over to the negative branch, or it may follow the positive branch and remain zero thereafter; this latter behavior can be assured by the introduction of stops. This type of linkage is therefore of value in mechanizing func- tions with a discontinuity in the derivative. Unfortunately, these cells cannot supply any appreciable effort near the point of singularity; Sec. 5-3] SINGULAR CASES OF THREE-BAR LINKAGES 115 torques must be applied to both cranks in the directions of the desired motions. In practical applications the author uses a still more special linkage, with Ai = Bi = A2 = B2 (Fig. 5-8). This is also a special case of the other singular classes, A, B, and C\\ it is interesting to observe how the diverse curves of Figs. 5-4 to 5-7 can pass over into the curves of Fig. 5-8 as a common limiting case. With this linkage three types of configura- Fig. 5-8. — Three-bar linkage with Ai = B\ B2. tion are possible, represented by three sets of lines on the graph in Fig. 5-8: o. The parallelogram linkage configuration, represented by the curves X2 = Xi + 2irn. b. Configurations in which the input terminal is locked in a definite position, Xi = (2n + 1)t, while the output terminal can assume any position. c. Configurations in which the output terminal is locked in a definite position, X2 = 2tu, while the input terminal can assume any position. Of particular interest are the transitions between configurations of types (b) and (c), which can be assured by the use of stops. We shall now see how these can be used in generating a function with a discontinuous derivative. Figure 5.9 shows a mechanical cell for which Xu = aXi Xk = bXi with when Xi > 0, when Xi < 0. (20) a > 1 > b. 116 THE THREE-BAR LINKAGE [Sec. 5-3 It consists of the linkage of Fig. 5-8, with added input and output termi- nals which are push-rods pivoted to the central link B2. The input and output parameters, Xt and Xk, are displacements of these rods perpendicular to the line of the pivots Si and S2 of the three-bar linkage. When Xi = Xft = 0, the linkage is in its critical position, with Xi = 180°, X2 = 0°; the two cranks then just touch stops Ch C2, which limit their motions to X1 ^ 180°, X2 ^ 0. If Xx is now increased by a push exerted on the Xi terminal, the crank A2 will be held firmly by the stop C2, while the crank Bx and the link B2 will rotate together about their col- linear pivotal axes Si and T2 into, for example, the configuration illustrated in Fig. 5-9. The param- eter Xk will then increase more rapidly than Xi, in the ratio of the distances of the corresponding push-rod pivots from the axis of rotation : 0 • 'S2 /b$ A2 Bw Ifafel xk^ iff / ) X; — f/f < if®}// s ( J p— ^c2:©| Fig. 5-9. — Mechanical cell generating a func- tion with discontinuous derivative. a = B, B, - d> (21) If the direction of motion is reversed by a push exerted on the Xk terminal, A 2 will be held against the stop C2 both by the linkage constraint and by the torque due to any resisting force at the Xi terminal, until the crank Bi touches the stop d. At this point the situation changes abruptly: the crank Bi can no longer rotate; the crank A2 is no longer locked in position by the linkage constraint; a further push on the Xk terminal will cause the crank A2 and the link B2 to rotate together about their now collinear pivotal axes S2 and 7Y Then Xi and Xk both become negative, the ratio of their values being b = ch (22) The change in dX2/dXi as the linkage is pushed through its critical posi- tion, in either direction, is quite abrupt; it is associated with a similarly sudden increase in the driving force necessary to overcome a resisting force at the other terminal, when the mechanical advantage is reduced by the change in fulcrum. The desired discontinuity in the derivative is not so perfectly realized if the link is pulled rather than pushed through its critical position. When the configuration is that illustrated in Fig. 5-9, a pull exerted on the Sec. 54] PROBLEM OF DESIGNING THREE-BAR LINKAGES 117 Xi-terminal and a resisting pull on the X^-terminal will produce a torque tending to move the arm A2 away from its stop. This arm, however, is locked in position by the linkage constraint, and the locking will be effective until the critical position is approached, and mechanical play in the linkage becomes important. This will allow crank A2 to begin to move away from stop C2 before crank B i- quite reaches stop C\; the result is some rounding off of the otherwise abrupt transition from one slope to another, but there is no tendency for the mechanism to jam. 54. The Problem of Designing Three -bar Linkages. — We have now to consider the problem of determining the elements of a three-bar linkage that will mechanize a given function x2 = (x2\xi) • xi. (23) If this function is to fall within the class of linkage functions, it must be required to generate it only for a finite range Azi of the input variable Xi, or, if the range of x\ is infinite, x2 must be a periodic function of Xi with period Aa?i. In either case, restricting attention to the range Azi of the input variable, one can write the relation in homogeneous form: fa = (fa\fa) ■ fa. (24) To mechanize this relation we have to design a three-bar linkage described by X2 = (X8|Xi) • Xi, Xlm ^ Xi ^ XlM, (25) such that when homogeneous parameters H1} H2 are introduced, the corresponding relation H2 = (H*\HX) • Ht (26) becomes identical with Eq. (24) on direct or complementary identifica- tion of the pair of variables (fa, fa) with the pair (Hi H2). If the func- tion to be generated is periodic, it is necessary, in addition, that AXi = Xim - Xln = 360°; the infinite range of Xi then corresponds to the infinite range of Xi, when passage to the next period of the generated function is permitted. A three-bar linkage may be described by the constants Ai Bi, A2, B2, Xim, Xim, AXi, X2m, X2M, AX2; of these only five are independent. The form of the function (X2|Xi) is determined by three independent ratios of the sides of the quadrilateral; the angles X\ and X2 do not depend on the over-all scale of the mechanism. We shall choose the three side-ratios, Bi/ 'Ah B2/A2, Ai/A2, as the independent constants that determine the form of (X2|Xi). Now, the field of functions (X2|Xi) 118 THE THREE-BAR LINKAGE [Sec. 5-5 of the three-bar linkage is three-dimensional, but each function (X2|Xi) can generate a whole field of functions (H*\Hi) that depend on the choice of additional constants: two constants (for example, X\m and X\M) in the case of a nonperiodic function, and one (for example, X2m) in the case of periodic functions. The field of all functions (H2\Hi) of a three- bar linkage is therefore five-dimensional where nonperiodic functions are concerned, and four-dimensional with respect to periodic functions. We shall henceforth concentrate our attention on the more difficult case of nonperiodic functions. In practical terms, the problem is that of approximating a given func- tion (/i2|/ii) as well as possible by a three-bar-linkage function (#2|#i) characterized by five independent constants. It is very difficult to find the best fit by varying all five constants independently; one must begin by assigning fixed values to at least two of them, even when choice of these values must be made rather arbitrarily. Fortunately, in prac- tice one has usually some indication of an appropriate value for one or more of these constants. The way in which a linkage is used in the computer as a rule suggests an appropriate value for AXi and AX2. In particular, in generating a monotonic function one can hardly have AXi > 180°; on the other hand, AXi must not be chosen too small lest the linkage degenerate into what is essentially a harmonic transformer. It is thus evident that it will be useful to have a method for finding the best fit to the given function consistent with specified values of AXi and AX2; the side-ratios (or their equivalent) will then be the adjustable parameters. The nomographic method, to be described immediately, is suited for this type of curve fitting. It should be used for all monotonic functions and is useful in many other cases. When the given function is not monotonic, it is sometimes difficult to choose AXi. The geometric method, to be described later in this chapter, is then useful. In applying this method, AX2 and B2/A2 are fixed and the fit to the given function is obtained by adjustment of AXi, Bi/Ah and Ai/A2. THE NOMOGRAPHIC METHOD The " nomographic method" here discussed is a method of curve fitting by three-bar linkages with given AX! and AX2. It takes its name from the use made of an intersection nomogram , which is reproduced following page 334 of this book. This nomogram, Fig. Bl, is also useful in many other types of calculations on three-bar linkages. 5*5. Analytic Basis of the Nomographic Method. — For analytic pur- poses it is convenient to specify the side-ratios of the quadrilateral through the three independent constants Sec. 5-5) ANALYTIC BASIS OF THE NOMOGRAPHIC METHOD 119 *>! = !« (J) (27) &2 = ln(f«), (28) a = ln(£), (29) Correspondingly, we may specify the configuration of the quadrilateral in terms of the diagonal-to-side ratio, through one or the other of the new variable parameters Pi = In (£). (30) p2 = In (£) = Pi + o, (31) which will replace the input parameter Xi in our discussion. In terms of these new symbols the equations of Sec. 51 take on a less familiar but very useful form. Since £-«*. $-* £-* 3-:**. ^ one can rewrite Eq. (2) as 62Pl = 1 + e2&1 + 2efti cos Xi, (33) or etPl = 2e\ (?L+J_!Ll + cos X\ (34) Hence the relation between the variable parameters Xi and pi is given by cos Xi = i e2p i-*i - cosh &i, (35) or pi = i In (2 cos Xi + 2 cosh bi) + i &i. (36) By similar manipulations Eqs. (3) and (4) become, respectively, cos 771 = cosh pi — ie26ie~pi, (37) cos T72 = cosh p2 — ie262e-p2 = cosh (pi + a) - ie»^-<*»+«). (38) As before, sin 171 and sin Xi must have this same sign, while 0 ^ 7?2 ^ 180°. Then the output parameter is given by X2+ = vi + i?«, (39) 120 THE THREE-BAR LINKAGE [Sec. 5-6 or by X2- = vi ~ n* (40) Equations (36) to (40) describe all three-bar-linkage functions (X2\Xi). The important feature of this formulation is the expression of rji and 772 in terms of the same function of two independent variables, G(p, b) = cos-1 (cosh p - ie2^); (41) one has Vi = G(pi, 61) (42) and V2 = Gfa, b2) = G(Pl + a, b2). (43) This makes it possible to compute 771 and 772 by the same intersection nomogram, with other advantages that will become clear as the discussion proceeds. 5*6. The Nomographic Chart. — In three-bar-linkage calculations one repeatedly encounters the relations V = G(p, b) = cos-1 (cosh p - ie2b~p) (44) and p = F(X,b) = i In (2 cos X + 2 cosh b) + i&, (45) where p stands for px or p2 = pi + a, b for 61 or b2, X for X1} and 77 for 771 or 772. It may be required to solve these equations singly or simul- taneously, with various choices of the unknown. For rapid calculations of this type the use of an intersection or grid nomogram is very convenient. The intersection nomogram representing a given functional relation is not uniquely determined, but may be given an infinite variety of forms. In the present case it is desirable to take lines of constant p as vertical lines, lines of constant 77 as horizontal lines, and to plot on the (p, 77)-plane curved lines of constant b and constant X (Fig. 5-10). It is at once evident that choice of consistent values of any two of the variables will determine a definite point on the chart — the intersection of the lines corresponding to the given values of these variables; corresponding values of the two other variables, as determined by Eqs. (44) and (45), can then be read off at the same point. Before illustrating this process, however, we must consider in more detail the structure and properties of the chart. As shown in Fig. 5-10, the horizontal scale is uniform in p with the vertical lines spaced at intervals of 0.1 In 10; they are labeled in terms of the variable (?) »P = logio I J h (46) for which the intervals are 0.1. The vertical scale is uniform in 77, with lines of constant 77 shown in Fig. 5-10 at intervals of 30°, from —180° to +180°. Sec. 5-6] THE NOMOGRAPHIC CHART 121 On the grid thus established there have been plotted lines of constant b at intervals of 0.1 In 10; they are labeled in terms of the variable jib = logio -r> (47) for which the intervals are 0.1. (The factor \l = 1/ln 10 is introduced in this way to facilitate computation with decimal logarithms.) The curve b = 0 is open, with the horizontal asymptotes rj = ± 90°. Curves + 180" -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 +0.1 +0.2 +0.3 +0.4 MP Fig. 5* 10. — Intersection, nomogram solving Eqs. (44) and (45). of constant b < 0 are closed. Curves of constant b > 0 are open and periodic in rj with period Arj = 360°; they have a pronounced sinusoidal character, being symmetric to reflection in the lines rj = • • • -180°, 0°, 180°, . . . and centrally symmetric about the points where they cross the lines v = • • • -90°, +90°, .... For a more detailed discussion the reader is referred to Appendix B. 122 THE THREE-BAR LINKAGE [Sec. 5-7 Lines of constant X have been plotted at intervals of 10°. All are open curves, and each has the same shape as a part of the curve 6 = 0. Indeed, the curves X = X0 and X = X0 — 180°, which join smoothly atp = ?7 = OifO<Xo< 180°, together form a curve congruent with the curve 6 = 0. All have asymptotes parallel to the p-axis, but run to infinity toward the right (p = +»), instead of toward the left (p = — oo ) as does the curve 6 = 0. Again the reader is referred to Appendix B for a more complete discussion. Since the parameters p and rj have no limits, the nomogram extends in principle over the whole plane. It is periodic in ij with period 360°; the part shown in Fig. 5-10 could be supplemented by the addition of similar figures above and below, extending indefinitely to positive and negative rj. The chart could also be extended to larger and smaller p, but the added portions would be of less practical importance since very large or very small values of 6 are not much used. In actual work one does not need the whole field covered by Fig. 5-10 but only its upper half, since the lower half is a mirror image. By sup- pressing the lower half, longer scales can be used in a given available space. This has been done in the preparation of Fig. B-l, which pre- sents this nomogram on the largest scale possible in this book. This figure is quite adequate for a study of the method ; in actual design work it is desirable to have it drawn on a scale four times as large and with a greater number of curves. Table B-l of Appendix B presents the informa- tion needed for redrawing the nomogram — the coordinates (jxP, \\) of the points of intersection of the curves /x6 = 0, ±0.01, ±0.02, • • • , ±0.5, with the curves X = 0, ±5°, ±10°, • • • , ±180°. 5*7. Calculation of the Function Generated by a Given Three -bar Linkage. — The intersection nomogram permits solution of Eqs. (36) to (40), which completely describe any three-bar linkage; it therefore suffices for the graphical construction of any three-bar-linkage function (X2|Xi). The procedure will be described in connection with its application to the special linkage sketched in Fig. 5-11, for which /x6i = —0.1, /x62 = 0.3, ixa = 0.3. For this linkage ^ = 0.795, f2 = 1.995, ^ = 1.995; with Jxi A.2 A.2 B2 taken as unity, the links have lengths B\ = 0.795, Ai = 1.000, A2 = 0.501. To determine the value of the output parameter X2 corresponding to a given value of Xi — in the example, 140°, as illustrated in Fig. 5-11 — we proceed as follows: (1) Knowing Xi and 6i, one can determine pi and 771 by Eqs. (36) and (37). Instead, on the nomogram, Fig. 5-12, we follow the curve X = Xi (=140°) until it intersects with the curve /x6 = /161 (= —0.1) at Sec. 5-7] FUNCTION GENERATED BY THREE-BAR LINKAGE 123 the point P(0). At this point we can read off the corresponding values of m ( = -0.191) and m (= 52.5°). (2) Knowing npi and fia, one can compute ju(Pi + a) — in the example 0.109. Instead, on the nomogram we locate a point fxa units to the right of P(0) (by the scale at the bottom of the figure) and through this con- struct the vertical line \xp = /x(pi + a). (3) Knowing n(pi + a) and /*62, one can compute 772 by Eq. (38). Instead, on the nomogram we follow the vertical line up = m(Pi + a) until it intersects the curve nb = ub2 ( = 0.3), as it does at the two points, Fig. 5*11. — Three-bar linkage used in illustrative calculations. Q(l> and Q(^, within the field of Fig. 5-12, and at an infinite sequence of points outside this field. It is at the point Q(+} that rj lies between 0 and 7r, and it is, therefore, at this point that we can read off the value of 772 (= 121°). (4) Computation of X2 is now simple: X2+ = 7?! + 772 = 52.5° + 121° = 173.5°, and X2_ = 771 - 772 = 52.5° - 121° = -68.5°. These values can be checked on Fig. 5-11. It will be noted that in Fig. 5-12 the value of 771 is represented by a vertical line from the line 77 = 0 to the point Pw, and the value of 772 is represented by a similar line to the point Q(£\ Graphical methods for adding these lengths can be used to construct the value of X2+. In the same way, the vertical line from 77 = 0 to the point Q{® represents the (negative) quantity which must be added to 771 to get X2-', we call this 7/2_, to distinguish it from the principal value, 772+, and write X2± = 771 -f- 772+. (48) 124 THE THREE-BAR LINKAGE [Sec. 5-7 We shall often use this relation instead of Eqs. (39) and (40). The point QJ? wul then be regarded as corresponding to the positive branch of the solution for X2, and Q*^ as corresponding to the negative branch. By use of the nomogram we can get a graphic presentation of the entire course of the function generated by a given linkage. To picture -0.7 -0.6 -0.5 -0.4 -0.3 -0.2-0.1 0.0 +0.1 +0.2 +0.3+0.4 HP Fig. 5-12. — Calculation of the function generated by a given three-bar linkage. the function (X2|Xi) we may wish to compute X2 for a " spectrum of values of Xi": Xf, Xi», X[2\ . . . X[n\ (In Fig. 512, X[r) = 140° - r-10°, r = 0, 1, • • • 4.) Corresponding to this sequence of values, there is a sequence of points P(0), P(1), . . . , on the curve pb = nbi, at which we can read off the spectra of values of HPi and of rji'. MP(i0), VP?, • • • M*; ni*, v?, . . . Vin)- Sec. 5-8] COMPLETE REPRESENTATION BY THE NOMOGRAM 125 Vertical lines from 77 = 0 to these points represent the spectrum of values of \ip\ by their horizontal spacing, the spectrum of values of 771 by their lengths. On shifting each of these lines to the right by an amount na we next obtain lines representing, by their horizontal spacing, the spectrum of the variable parameter fxp2 = n(pi + a), which can assume the values Kpf + a), KP? + a), • • • M(p(in) + a). Since this shift does not disturb the distribution of the lines, one can speak of the spectrum of values of np2 as congruent to the spectrum of values of npi. The spectral lines for jup2, by their intersections with the curve yb = /i62, define two sequences of points: Qf, «?, . . . QVn>, and QL0), Q(l\ . . . Qlw), from which one may read off the spectral values of 772: *»(0) -»(1) T7(n) By terminating the spectral lines of MP2 at the points Q(±\ we can make them represent the spectral values of 77 g by their upward and downward extensions, just as the spectral lines for npi represent the values of 771. There results a very clear picture of the way in which 771 and 772+ change together with Xi. Finally, the spectrum of values of the output variable, yd) yd) Y(n> can be obtained by adding corresponding spectral values of 771 and 772+. 5*8. Complete Representation of Three -bar-linkage Functions by the Nomogram. — It will soon become evident to the reader who attempts to use the nomogram that it is not possible to carry through for all Xi, and for given phi, nbz, and pa, the calculation outlined in Sec. 5-7. This limitation corresponds to restrictions on Xi inherent in the geometry of the linkage considered, and is not a shortcoming of the nomogram ; that the nomogram gives a complete representation of the whole class of three-bar-linkage functions will be evident from the following discussion. In the calculations discussed in Sec. 5-7 it is convenient, but not neces- sary, to select values of X\ corresponding to lines appearing on the nomo- gram. We shall now consider a continuous spectrum X({\ which includes all values in the range — 00 < X\ < + 00 . We shall call such a con- tinuous and infinite spectrum of X({} "the complete spectrum X?'." Corresponding to a continuous spectrum X^ there will be a continuous spectrum X({\ The values of X2r), however, as computed by Eqs. (36) to (40), will not be real for the complete spectrum X({} but only for certain "bands" of that spectrum. Real configurations of the linkage 126 THE THREE-BAR LINKAGE Jec. 5-8 correspond, of course, only to real values of X2r); thus, by observing the limiting values of X^ and X2r) in the " bands" in which the solution is real, one might determine the limiting configurations of the linkage. We now use the nomogram in studying the conditions for the existence of a real solution X2r) corresponding to a given value of X{{\ We note first that, as X(^ goes through all values, npi can go only through a limited range of values determined by the fixed value of ju&i- This cor- responds to the limitation on the magnitude of the diagonal D, which has Complete spectrum r Complete spectrum np^[ Limited spectrum UPp) pa Limited spectrum ppfi Fig. 5-13. — Range of operation of a three-bar linkage. been expressed analytically in Eq. (5) and in our present notation can be rewritten as logio |1 - 10^| ^ fiPl £ logib (1 + 10*0. (49) The range of np\ is finite if yb\ ^ 0, and extends to — « when ybi = 0; we shall speak of the values in the range of npi as making up "the com- plete spectrum /zPir)-" In Fig. 5-13, which applies to a linkage with ixbi = -0.2, fxb2 = 0.3, ixa = 0.5, it is clear that /*Pi can not be less than —0.432 (for Xi = 180°), nor greater than 0.215 (for Xi = 0°). We have, then, for the complete spectrum, —0.432 ^ np[r) ^ 0.215. By shifting the complete spectrum Sec. 5-9] RESTATEMENT OF THE DESIGN PROBLEM 127 /xp(!r) to the right by a distance pa, we obtain the complete spectrum /*(pir) + °)- In our example +0.068 ^ /z(pir) + a) ^ 0.715. In the same way it will be observed on the nomogram that there is a limited range of values of /zp2 consistent with the fixed value of ju&2. This corresponds to the restriction on D expressed analytically by Eq. (6). which in our present notation is logic |1 - 10"M ^ M2>2 ^ logio (1 + 10*.). (50) The values in this range make up the complete spectrum np({\ In the case illustrated in Fig. 5-13, -0.002 ^ MP(r) ^ 0.476. In the nomographic computation of X2 one has to identify /zp2 and m(Pi + <*>)• This will be possible only for values of y.pi which lie in the complete spectrum ^p(2r) and also in the complete spectrum ju(pir) + o); such values make up the ''limited spectrum np(2r)." By shifting the limited spectrum np2r) to the left by an amount pa, we obtain finally the limited spectrum fip({\ The nomographic computation can be carried through for all values of \ip\ that He within this limited spectrum; for the corresponding values of Xh the limited spectrum X{{\ one can com- pute real values of X2. The range within which this calculation is pos- sible corresponds exactly to the range within which both Eqs. (5) and (6) are satisfied, as illustrated in Fig. 5-2. Thus all physically possible configurations of the linkage, all real three-bar-linkage functions (X2|Xi), are covered by the nomogram. The reader will find it instructive to apply the nomogram to the dis- cussion of the parallelogram linkage. 5*9. Restatement of the Design Problem for the Nomographic Method. — The nomogram is conveniently used in three-bar-linkage design only when it is possible to preassign values for two of the design constants, AXi and AX2. There remain three design constants — 61, 62; and a, or their equivalents — to be adjusted in the process of fitting the generated to the given function. When the angular ranges of the input and output variables are thus specified, it becomes possible to express the given function in terms of angular variables (pi and <p2, instead of the homogeneous variables hi and /i2: <Pi = <P2 = AX2h2. j (51) In comparing the given function with the generated function, one will correspondingly express the latter in terms of the angular parameters Xi and Xq' 128 THE THREE-BAR LINKAGE [Sec. 510 (52) Xi — Xim = AXiHi, X2 — Xim = AX2H2. The design problem can then be stated as follows. It is desired to find a three-bar linkage generating a function X2 = (X2\X1) - X, (25) which can be identified with the given function <P2 = (<pi\'<pi) ' <pi (53) on direct or complementary identification of Hi with hi, H2 with h2 (cf. Sec. 54). Direct identification in the two cases implies (fl = Xi — Xlm, I xg^v <P2 = X2 — X2m\ ) complementary identification implies AXi — <P\ = X\ — Xlm, I ,_.,. AX 2 — <P2 = X2 — X2m. j The design problem is essentially the same if the identification is direct in both cases or complementary in both cases; if the identification is direct in one case and complementary in the other it does not matter in which case it is direct. It will be convenient to assume that it is always direct in the case of the output variable. The relations to be satisfied by the angular parameters are then ±<Pi = Xi- Xim - JAXi ± iAXx, j (56) v?2 = X2 — X2m, J with the upper sign corresponding to direct identification. It is important to note that the procedure to be described does not necessarily lead to a unique solution of the problem. There usually exist two quite different approximate solutions, with a positive and a negative value for 61, respectively. During the design process the constants of both of these solutions should be determined sufficiently accurately to permit a rational choice between them. This point will be fully illus- trated in later sections. 5-10. Survey of the Nomographic Method. — Fitting the generated to the given function by simultaneous and independent variations of the three remaining design constants is hardly practicable. We therefore (1) make a definite choice of 61, and then find the best fit obtainable by independent variation of the other two design constants; (2) find a better value of 61, as described in Sec. 5-13; (3) find the best fit obtainable by variation of the other design constants, using this improved value of 61; Sec. 510] SURVEY OF THE NOMOGRAPHIC METHOD 129 (4) find a better value of 6i; and so on, approaching the optimum choice of all three constants by successive approximations. It would be quite natural to choose b% and a as the design constants to be adjusted in the first step of this procedure. However, to deal with these constants directly involves, in effect, the fitting of the given curve to a member of a two-parameter family of three-bar-linkage curves. It is preferable to choose Xim and X2m as the additional constants on which attention is concentrated, since it is then possible to work instead with two one-parameter families of curves, one easily constructed from the given function, the other appearing on the intersection nomogram. To make it clear how this can be done we shall consider three increasingly difficult problems. The discussion will be illustrated by Fig. 5*14. -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 o'.O +01 +0.2 +03 +04 +0.'5 +0.6 MP Fig. 5-14. — Curve fitting in Problems 1 and 2, Sec. 5-10. Problem 1: Determine whether or not a given function (<p2\(pi) is generated by a linkage of specified constants AXi, AX2, 6i, Xim, X2m. — Since AXi, Xim, and X2m are known, it is possible to transform the given functional relation ((pz\<pi) into a functional relation of the angles Xi and X2, (X2IX1), by use of Eqs. (56), whether the identification is direct or com- plementary. The problem is then: Is this function (X2|Xi) really generated by a linkage characterized by the given constants? Let X(!r) be any value of Xi in the specified range. On the nomogram we locate at the intersection of the lines X = X{r) and fib = fibi the point P(r) ; at this point we can read off the values of np{r) and rj{r) generated by the linkage. We do not yet know the generated values of t?2± and X2, but we do know the corresponding given value of X2, X$, and can compute the "given" value of rj2± using Eq. (48): « = *B - rir)- (57) 130 THE THREE-BAR LINKAGE [Sec. 510 On the vertical line through the point P(r) we erect a spectrum line of height r)$, ending at point R(r). By moving this spectrum line to the right by an amount \xa (still unknown) it can be brought into the position of the spectrum line MP2r)- If the given value rfft is the one actually generated, this spectral line will then extend exactly to the curve \ib = nbz, (also unknown at the moment) ; the amount by which it falls short of that curve is the amount by which the generated value of X(2r) exceeds the given X%. The complete spectrum of such lines, limited by the curve R<-0) . . . R(r) . . . Rw, can be outlined quickly. Let the curve Rw . . . R(r) . . . R{n) be drawn on a transparent overlay. Suppose now that, by moving the overlay to the right by a distance i*a, this curve can be made to coincide with some curve fib = ju&2. It will then follow that the given function is indeed generated by a linkage with the specified constants, AXi, AX2, 61, Ximj X2m, and, furthermore, that this linkage is also characterized by the constants 62 and a thus determined. For, in view of the methods of computation outlined in Sec. 5-7, it is clear that a linkage with the above values of 61, 62, and a will generate a spectrum of values of 772+ which is just the spectrum of the "given" values, ijg}, and hence a spectrum of values of X2 which is also the given spectrum X%, for Xim ^ X ^ X\m + AXi. The spectrum of Xi is determined by the given constants X\m and AXi; the generated spectrum of X2, which reproduces the given spectrum of X2, must correspond to the constants X2m and AX2. Since the linkage that generates the given function is characterized by the five specified constants and by the derived 62 and a, the truth of the statement follows. As an example, shown in Fig. 5-14, we take a case in which the specified constants are AXi = 60°, AX2 = 105°, lxbl = -0.2, Xlm = 90°, X2m = 45°. The given function (<P2\<pi) has been assumed to increase monotonically from <p2 = 0° when <pi = 0° to <p2 = 105° when <p\ = 60°; by Eq. (56) (with the upper sign) X2 then increases monotonically from X2 = 45° when Xi = 90° to X2 = 150° when Xi = 150°. Taking X[0) = 45°, we locate P(0) and read r\f = 33°; hence ijg is 45° - 33° = 12°, corresponding to point Rw. With Xiw) = 150°, we locate P{n) and read ^ = 36°; hence t?2? = 150° - 36° = 114°, corresponding to point .R(n). We shall assume that similar computations for intermediate Xi serve to determine the curve R{0) . . . R(r) . . . Rin) as shown. If now this curve is moved to the right by an amount /xa = 0.4, the end points can be brought to lie at points A and C on the same contour of constant fib: fib* = 0.3. The intermediate portions of the curve do not then lie on that contour, and it is evident that no other contour can give a fit. It follows that the given function can not be generated by a linkage with the specified constants. The difference between the given and the generated Sec. 510] SURVEY OF THE NOMOGRAPHIC METHOD 131 functions is immediately evident. A linkage with pa = 0.4, pb2 = 0.3 does give a fit at the very ends of the range of Xi (points A and C), and thus at the ends of the range of X2. This linkage has therefore the specified values of AXi, AX2, Xim, and X2m, as well as that of &ij it is the specified linkage. It generates values of r)2± given by the curve AA'B'C, instead of the "given" values of the curve ABC; the vertical separation between these curves is then the difference between the given and the generated values of X2. Problem 2: Determine whether or not a given function ((p2\<pi) can be generated by a linkage of specified constants AXi, AX2, 61, X\m. — It is now possible to consider all values of X2m in seeking a fit of the generated to the given curve, instead of only one value. Let the curve Rw . . . R(n) be constructed as before, for an arbitrarily chosen value of X2m — for example, X'2m. If a fit can not be found for this among the curves of constant pb on the nomogram, one will desire to make a similar trial for another value of X2m — for example, X'L = X'2m + A. (58) By Eq. (56), this increase in X2m will produce a uniform increase, by the same amount, in the "given" values X2g and rj2g; the new curve Rw . . . R(n) will be the old one raised by an amount A, and the fit will be sought as before. Of course, instead of redrawing the curve, one can simply shift upward by A the overlay on which the first curve was drawn. Thus by allowing all vertical shifts of the overlay in seeking a fit one treats X2m as a disposable parameter. The stated problem can then be solved as follows: On a transparent overlay draw a curve Rw . . . R(r) . . . R{n), assuming X2m = X'2m. If, by translating the overlay to the right by an amount pa and upward by an amount A (as read on the scale of rj), this curve can be made to coincide with some portion of the curve pb = pb2) then the given function (^2|^>i) can be generated by a linkage with the specified constants AXi, AX2, 61, and X\m. Furthermore, that linkage will also be characterized by the constants 62, a, and X2m = X2m + A, determined in this fitting process. In the example of Fig. 5-14 a fit can be obtained by moving the over- lay, prepared as previously described, upward by an amount A = 57°, and to the right by pa = 0.255; curve Ri0) . . . R{n) then lies on the curve pb = pb2 = 0.3, extending from A' to C. Thus the given curve is actually generated by a linkage with the constants AXi = 60°, AX2 = 105°, pbi = -0.2, Xlm = 90°, and, as now determined, pb2 = 0.3, pa = 0.255, X2m = 45° + 57° = 102°. If the fit were not exact the difference between the generated and the 132 THE THREE-BAR LINKAGE [Sec. 5- 11 given functions could be read as the vertical separation of the curve #(0) . . . R{n) and the contour pb = yb%. Problem 3: Determine whether or not a given function (^>2|vi) can be generated by a linkage of specified constants AXi, AX2, 61. — Both X\m and X<Lm are now to be treated as disposable constants; the problem is then essentially the same as that encountered in Step (1) of the design pro- cedure described at the beginning of this section. We have already seen how a fit can be sought for a given value of Xim — for example, X'lm — by a process that begins with the construction of a corresponding curve — for example, R(o0) . . . R(on)- To make the same test for another value of X\m — for example X"m — one would similarly construct another curve, R(r0) . . . R[n). Unfortunately, this is not of the same form as the first curve; the actual construction of this curve is not to be avoided, though it can be made relatively easy by methods to be described. The problem is then to be solved as follows. On a transparent over- lay, construct a family of curves R(0) . . . R<-n) for sufficiently closely spaced values of X\mj and for X2m = 0; label each curve with the corre- sponding value of Xim. Now suppose that, by translating the overlay to the right by an amount \ia and upward by an amount X2w (as read on the scale of 77), the curve of this family labeled X\m can be brought into coincidence with a part of the curve ixb = /x62 on the nomogram. Then the given function can be generated by a linkage with the given con- stants AXi, AX2, and bi'} this linkage would be characterized also by the constants Xim, X2m, jua, pb% thus determined. The essential features of the nomographic method should now be evident to the reader. To find the three-bar linkage with given AXi, AX2, 61, which most accurately generates a given function (<p2|^>i), one constructs on an overlay a family of curves corresponding to X2m = 0 and to various values of X\m. Moving the overlay over the nomogram, one seeks the best possible fit of a curve of this family to a curve of the ju&2 family on the nomogram. The displacement of the overlay necessary to produce this fit determines X2m and /xo for the linkage; the choice of curves for this fit determines X\m and /zb2. The error in the resultant mechanization is directly evident in the failure to obtain an exact fit between the overlay and nomogram curves, and is measured by their vertical separation. The steps involved in this process will be discussed in detail in Sees. 5-11 and 5-12. After the method of improving the choice of 61 has been described in Sec. 5-13, the whole procedure will be fully illustrated in Sec. 5-14. 5- 11. Adjustment of b2 and a, for Fixed AXi, AX2, b\. — We shall now describe in full detail a practical procedure for the construction of the Sec. 5.11] ADJUSTMENT OF B2 AND A 133 overlay mentioned in Sec. 5-10 and its use in determining the best values of 62 and a. Construction of the Overlay. (1) Choose a spectrum of values of Xi, X[s) = s8, (59) which fills the entire range from 0 to 360° at intervals 8 small compared to AXi. One should choose 8 as the difference in X between consecutive curves on the nomogram, or a multiple of this, so that there will be on the chart a curve corresponding to each value X\s). Usually 5 = 10° is sufficiently small; 8 = 5° is possible with the chart plotted from Table B-l. (2) As the spectrum of values of <pi, take rf* = r8, (60) with r — 0, 1, • • • n. Since these values should fill the range AXi, one must have n8 ^ AXi. (3) Compute the corresponding spectrum of <p2: <P? = Mvi) ' <p[r). (61) Using the same scale as the 77-scale of the nomogram, construct this spectrum as a series of tiny holes along a straight line on a strip of paper (see Fig. 5-15). On this strip mark the index r for each of the lines of the spectrum ; indicate by an arrow the direction of increasing <pi. (4) Fasten over the nomogram the material on which the overlay is to be constructed — for instance, a piece of tracing paper. Copy onto the overlay all the points P(s) at which the contour ju6 = yb\ is inter- sected by the lines X = X[s). (Figure 5-15 shows the complete contour.) Mark the points P(s) on the overlay with the subscript s. Also copy onto the overlay the lines t\ = 0, p = 0. This position of the overlay will be called its starting position. (5) Draw on the overlay the vertical lines jip = jup(s) through all points P(s). These are the spectral lines for the variable up. The over- lay can now be separated from the nomogram. (6) Place the strip of paper carrying the spectrum (p(2r) on the overlay, along each line of the spectrum npis), making the arrow point downward and the first point <p(20) of this spectrum coincide with the point P(s). For each such position of the strip, mark on the line ixp = np{s) of the overlay the positions of the points (p(2r) on the strip, labeling each with the corre- sponding value of r. These points we shall indicate as p(s)D(s) p(s) p(s) mm . . . rr ... rn . (7) Starting at each point P(s) on the overlay, pass a curve succes- sively through the points P(s), P[s+1), P(2s+2),. . . P<.s+r), . . . P<?+w). 134 THE THREE-BAR LINKAGE [Sec. 511 This family of curves we shall call the plus family. It is unnecessary to use a French curve in this construction; it is sufficient to connect the points by hand with straight lines, in order to make clear the way in *=*18 Fig. 5-15. — Scale and overlay for first application of the nomographic method. which they are associated. The plus family of curves appears as con- tinuous lines in Fig. 5-15. (8) Again, starting with each point P(s) on the overlay, pass a curve successively through the points P(*\ P^1', P(2*-2), . . . P(rs~r\ . . . P£-n). This family of curves (dashed lines in Fig. 5-15) we shall call the minus family. This completes the construction of the overlay. As will appear from our later example, construction of the complete overlay is not always necessary. We must now examine the significance of the curves thus drawn. (1) The successive points P(8), P<s+1>, . . . p(*+»\ represent, by their distances from the lines p = 0 and 77 = 0, the values of up and 771 for a sequence of values of Xi: X?}, Z(]8+1,) ... By Eq. (54) these points also correspond to the sequence of values of <pi". 0, (p\h . . . <p\n\ for the case Sec. 5-11] ADJUSTMENT OF B2 AND A 135 I in which X\m = X(8) and the identification of hi and Hi is direct. In particular, P<8+r> represents the values of np{r) and rff when Xlm = X{8\ (2) The separation of the points on the strip corresponding to <p(20) and <p(2r) represents, on the same scale, the value of <p(2r) — <p20) = <p2r). By Eq. (54) this is also the value of X2r)g if X2m = 0, and the identification of hi and Hi is direct. (3) The point P(r8+r) thus corresponds to \ip = /xp(r) and has as its t\ coordinate [by Eq. (57)] Vl ^1g — rl2g, for the case in which X\m = X({\ X2m = 0, and the identification of hi and Hi is direct. (4) We thus see that the curve of the plus family passed through the points P(8), P[8+l), . . . P(rs+r), . . . P(n+n) on the overlay corresponds to the sequences of values Xi: X{°\ X^», . . . X[8+r\ . . . X[8+n\ f*p: (xp(0), MP(1), . . • MP(r\ ■ • • MP(n)> >?• ^2»j ^20? • • • ^2»> • • • V2g) for the case described under (3). (5) The sign of rj can be reversed by rotating the figure through 180° about the axis 77 = 0. Thus if the overlay as constructed is turned face down by rotating it through 180° about the axis ij = 0, then the curve of the plus family labeled with an 5 gives the relation between rj2g and fip for the case in which X\m = X^, X2m = 0, and the identification of hi and Hi is direct. (6) In the same way the reader will be able to show that, with the over- lay turned face down as above, the curve of the minus family labeled with an s gives the relation between rj2ff and \ip for the case in which X\m = X^, X2m = 0, and the identification of hi and Hi is complementary. Use of the Overlay. — With the overlay face down on the nomogram and the line 7] = 0 horizontal, a fit is sought between any curve on the overlay and a line ^b = ixb2 of the nomogram. If a fit is found, the constants of the linkage are determined as follows: (1) AXi, AX2, ubi have been previously chosen. (2) fib2 is read from the curve of the nomogram with which the fit is found. (3) na is the shift of the overlay to the right needed to establish the fit. It may be read at the intersection of the line p = 0 of the overlay with the /zp-scale on the nomogram. 136 THE THREE-BAR LINKAGES [Sec. 512 (4) Xzm is the shift of the overlay upward needed to establish the fit. It may be read at the intersection of the line t\ — 0 of the overlay with the 77-scale of the nomogram. (5) If the fit is established in the plus family of curves, one has Xim = s8, s being read from the overlay curve with which the fit is made. The angles <pi and Xi will increase together; this, of course, is always true of <p2 and X2. (6) If the fit is established in the minus family of curves, one has Xim = s8, Xim = s8 — AX 1. The angle <pi will decrease as Xi increases. The linkage and the associated scales are then completely determined. The actually generated values of 172 will not be iffi, since these were computed on the assumption that X2m = 0; instead, they will be which can be read directly on the nomogram scale. If the fit is estab- lished on the upper half of the nomogram, 772 is greater than zero and the generated function belongs to the positive branch; if the fit is established on the lower half of the nomogram one has to do with a negative branch. Usually the fit obtained between the nomogram and overlay curves will be only approximate. The constants determined alone will then not all be mutually consistent, unless the approximate fit is so made that the error vanishes at the extreme values of X\ and X2. This is easily done when monotonic functions are being dealt with; in other cases one should remember that when five of the design constants have been fixed the others must be determined by appropriate calculations rather than read as above. 5*12. Alternative Methods for Overlay Construction. — Modifications of the above procedure are necessary when use is made of a nomogram like Fig. B-l, which includes only the range from 7? = 0° to r\ = 180°. The missing portions of this chart can be constructed as mirror images of the part shown; or, more conveniently, the same effect can be obtained by appropriately turning the overlay. For example: To construct the points P(s) on the overlay, copy the points P(0) to P(18) (assuming 8 = 10°) from the nomogram, and draw the reference lines. Then turn the overlay face down by rotating it about the line 77 = 0, and copy the same points onto the overlay. The points thus constructed are in fact P<0), P(-1}, P<"2), . . . p(-«>, and should be so labeled. In the curve-fitting process described above, with the overlay face down, the lower part of the nomogram will be missing, and direct fitting to functions of the negative branch will not be possible. One can, how- ever, turn the overlay again (so that it is now face up) and seek a fit on Sec. 513] CHOICE OF BEST VALUE OF 61 137 the upper part of the nomogram. It must, of course, be remembered that readings made on the rj-scale (for instance, readings of X2m) must then be taken with a minus sign. When ju&i > 0, an overlay constructed as described above becomes excessively large; another modification in the overlay construction then becomes convenient. It will be noted that if the lower half of an overlay is turned about the line -q = 0, the point P(0_s) will be brought into coincidence with the point P{0S\ and the point P(~s) will lie as far above P{08) as P(r8) lies below it. We shall speak of these points in their new posi- tion as the " transferred points"; they extend through the " transferred region" of the overlay. These transferred points can be constructed directly by the method described above with the one change that, in locating the transferred points Po_s), P(fs\ . . . P»~s), one places the point <?20) of the spectrum strip on the point P(0S) with the arrow directed upward before copying off the succession of points v?(20), <pi>l), . . . <pf\ In working with this transferred region one must remember that it is equivalent to a normal region turned face down. When fitting curves in a normal region, one turns the overlay face down and reads values directly from the 77-scale of the nomogram; when fitting curves in the transferred region, one uses the overlay face up. The plus and minus families of curves in the transferred region are most readily identified by turning over the overlay. 5'13. Choice of Best Value of bi for Given AXi, AX2. — In the preced- ing sections we have seen how to find the elements of a three-bar linkage which gives an approximate mechanization of a given relation, <P2 = (<P2\<Pl) ' <Pl, (53) when AXi, AX2, and fibi are specified in advance. We have now to con- sider the problem of finding an appropriate value for 61 when only AXi and AX 2 are specified. A method of trial and error is obviously applicable. One can carry through the above process for an arbitrarily chosen ju&ij if an acceptable fit is not found another value can be chosen for fibi and the process repeated, until a good fit is found or the useful range of fibi has been covered. Fortunately it is necessary to try only a relatively small number of values, such as ju&i = —0.5, —0.2, 0.0, 0.2, 0.5, to determine roughly the value of fibi or to establish that the proposed type of mecha- nization is not appropriate. Such a process of repeated trials can be abandoned as soon as even a poor approximate fit is found between the overlay and nomogram curves. Usually one finds at least a very rough fit with the first chosen value of lib\, and can begin to apply a second method — one of successive approxi- 138 THE THREE-BAR LINKAGE [Sec. 513 mations. Let the linkage that gives the first rough fit be characterized by the constants AXh AZ2, pip, »b%\ na^, (62) of which the last two have been found by the process already described. Now let us consider the problem of similarly mechanizing the inverted function <Pi = (<Pl\<p2) ' <P2, (63) with <p2 playing the role of the input parameter, <pi the role of the ouput parameter. The parameters <pi and <pi will then be interchanged through- out the previous discussion, <p± varying with the angle X2, <pi with the angle Xi. The linkage that mechanizes this relation will be the same as that which mechanizes the original relation, Eq. (54), except that input and output are interchanged. If Fig. 5-1 represents the linkage for Eq. (54), the linkage for Eq. (63) can be obtained from this by mirroring it in a vertical line, along with the associated scales for <pi and <p2. This new linkage differs from the old in that Bi and A 2 are interchanged, as are AXi and AX2; Xlm is replaced by 180° - X2M, X2m by 180° - X1M. As for the constants ybi, /x62, na, we note that interchange of Bx and^2 carries na = logio -^ (64) into logio -g1 = - logio -^ = — m&i, (65) and conversely, while becomes M&2 = log10|? (66) logl0(f)=logl„[(j)(^)(^)]=^-,a-,6, (67) Distinguishing the constants of the inverted linkage by a tilde, we may write ixa = — ubi, (68a) /i5i = — pa, (686) m62 = ju62 — /ia — /x6i, (68c) AXX = AX 2, (68d) and so on. In attempting to mechanize Eq. (63) one might apply the nomographic Sec. 514] AN EXAMPLE OF THE NOMOGRAPHIC METHOD 139 method as before, choosing arbitrarily a value of ju6i and finding corre- sponding values of nb2 and /xa. However, — fxa(1) is a known first approxi- mation to the desired value of /x6i, and an appropriate choice for the fixed value of this quantity. We therefore take plf = -fxa^, (69) and by the nomographic method determine the corresponding constants nb22) na(2) in the mechanization of the inverted problem. This mecha- nization of the relation between <pi and <p2 must be at least as good as that described by the constants in Eq. (62), since it is chosen as the best of a family of linkages which includes the mirror image of that first linkage; usually it is much better. From these constants one can then obtain second approximations to the constants required for the direct problem : fia^ = -nbf = tia(1), fib[2) = -mS(2), (70) fjLb(22) = >ibf - M5(2) - »b?\ The values of /xbi and nb2 have been improved; the value of fia was frozen in passing to the inverted problem, and is hence unchanged. We can now return to a consideration of the problem as first formu- lated. It is obviously desirable to take — /xd(2) = yb22) as the chosen value of fxbi; repetition of the curve-fitting process leads to a still better mechanization of the relation between <pi and <p2, characterized by the constants /*&$*>(= -m«(2)), M&(23), Ma(3). Thus by alternately considering the problem as formulated in Eqs. (53) and (63), and applying the methods of Sees. 5-11 and 5-12, one obtains successively better approximate solutions, which usually con- verge rapidly to a limit. The method is less laborious than might at first be supposed, since the constants to be expected in all solutions but the first are known approximately, and the complete overlay need not be constructed. It will be found that if one obtains a fair fit with a given /x&i, one will obtain also a reasonably good fit with — ju&i. On application of the method of successive approximations, these two approximate solutions usually lead to two different solutions of the problem, which are equivalent neither with respect to the residual error, nor with respect to mechanical qualities. These two possibilities should receive separate consideration. 5»14. An Example of the Nomographic Method. — As a first example of the nomographic method, we shall apply it in attempting to mechanize the function presented in Table 5-1 in both direct and inverted forms. 140 THE THREE-BAR LINKAGE Table 51. — Given Functions for the Example [Sec. 5- 14 (ftjft) (ft ft) ft, ft, fti ft, degrees degrees degrees degrees 0.0 0 0.0 0 22.3 10 3.4 10 34.1 20 9.0 20 43.6 30 16.9 30 52.1 40 26.6 40 60.0 50 37.7 50 68.3 60 50.0 60 75.9 70 62.6 70 83.1 80 76.1 80 90.0 90 90.0 90 This tabulated function is in fact the one generated by a three-bar linkage with the following constants: Xlm = -170°, AXi = 90°, X2m = 160°, AX2 = 90°, ubi = 0, ixa = -0.286, \xh2 = 0.0367, A1 ' A1 = 1.932, 4i = 2.102. (71) In applying the nomographic method we shall assume the ideal values for the angular travels, AXi = AX2 = 90°, (72) but shall begin by choosing a value of pb\ which is not the best: M&i» = -0.1. (73) In this way we can make particularly evident the convergence toward the best constants that is usually afforded by the method of successive approximations. A second and quite different example, without this ad hoc character, will be found in Sec. 6-4. Following the steps outlined in Sec. 5-11, we proceed thus: (1), (2) In tabulating the given function, 8 = 10° has been chosen as sufficiently small compared to the ranges of Xi and X2; this will permit use of Fig. B-l (folding insert in back of book) in applying the method. In mechanizing the function in the direct form the spectrum of values of <Pi is 0°, 10°, . . . 90°. We have here n = 9. (3) The values of <p2r) appear in the first column of Table 5-1. Using the ij-scale of the nomogram, we transfer this spectrum of values to a strip Sec. 514] AN EXAMPLE OF THE NOMOGRAPHIC METHOD 141 of paper, as illustrated in Fig. 5-15. The direction of increasing r is shown by an arrow. (4) Tracing paper is used in making an overlay. This is taped to the nomographic chart, which should be made on cardboard, and 36 points, from P<-18) to P(+18\ are constructed and marked with the proper value of s (Fig. 5-15). The reference lines are traced onto the overlay. (5) The vertical lines of the spectrum jup(s) are omitted from Fig. 5-15 for the sake of clarity. (6) Placing the zero point of the strip successively on each of the points P(s), with the arrow directed downwards, the 36 points PJ° are located and labeled with their r- values. (In first approximations one can sometimes skip half the values of r and half the values of s.) (7), (8) The plus family of curves is now sketched (full lines in Fig. 5-15) through points with r- values successively increasing by 1 as s increases; curves of the minus family (dashed in Fig. 5-15) pass through points with r-values successively increasing by 1 as s decreases. The complete family of curves is shown in the figure. This is really unneces- sary, since one can tell at a glance that some of them cannot lead to a fit. In particular, since <p2s) is a single-valued function of s, one could here omit the numerous curves that have infinities in their slopes. The overlay is now turned about the horizontal reference line and translated over the nomogram until a fit is found — a quite satisfactory fit, as it happens, between the overlay curve s — — 16 of the plus family and the curve yb2 = 0.075 on the nomogram. Figure 5-16 shows, on the nomogram grid, the construction of the particular overlay curve for which the fit was obtained, and the position of fit on the chart (dotted curve at lower left) . The fit has been made exact at the ends. The overlay curve then deviates downward from the nomogram curve; on a large chart it can be seen that the maximum error in -q is a little more than one degree. The reference lines on the overlay are also shown in the position of fit. The elements of the linkage are thus established: ph1™ = —0.1, as assumed. nb^ = 0.075, read from the nomogram curve on which the fit was made. jiia(1) = —0.265, read at the intersection of the vertical reference line with the jup-scale. X2m = —202.5° or +157.5°, read at the intersection of the horizontal reference line with the 77-scale. (When this reference line falls off the nomogram, as it would here, an auxiliary reference line on the overlay can be used.) X^ — —160° = s8, since the curve that gives the fit is of the plus family. 142 THE THREE-BAR LINKAGE [Sec. 5-14 By Eq. (56) we have (using the upper signs in the first equation, since the fit was obtained with" a curve of the plus family) * = Xx + 160°, <P2 = X2 — 157.5°. (74) These last equations represent the given function with errors visible as the vertical separation of the fitting curves in Fig. 5-16. Since the fit is exact 0.7 -0.6 Fig. 5-16. — Construction of overlay line and position of fit in first application of the nomographic method. In the position of fit the overlay curve lies slightly below the contour fj) = 0.075. at the ends of the range of X\, the travels of both input and output of the linkage as designed will have the required value, 90°. (It is not necessary to make the fit exact at the ends, except perhaps in the last stage of the design process. In earlier stages one can often accelerate convergence on the ideal constants by seeking a good fit on the average rather than an exact fit at any given points in the range.) As a check it is useful to make a drawing of the linkage, showing the cranks in their extreme positions (Fig. 5-17). The distance A[X) between the crank pivots may be taken as the unit of length; the relative crank lengths are drawn in as IP A[v = 10i*i<y = 10-0-1 = 0.794, = 10-"«(1) = 100-265 = 1.841. (75) Sec. 5-14] AN EXAMPLE OF THE NOMOGRAPHIC METHOD The constancy of the required length of the connecting link, A{» j±2_ _ ^2_ . fM- = i0"(6,cl)-°a)> = 100-340 = 2.188 A? A™ 143 (76) provides a check on the quantities determined in the fitting process. Of the constants thus determined for the linkage, 6i has been held at a preassigned value, but the others have taken on values that are good approximations to those known to give an exact fit. Such behavior is of course essential if the method of successive approximations (Sec. 5-13) is to be effective. We now apply this method to the improvement of the linkage design. Fig. 5-17. — First approximate linkage for mechanization of the given function, Table 5-1. The roles of <pi and <p2 are to be interchanged throughout our next treatment of the problem. The inverted function has already been given in Columns 3 and 4 of Table 5*1. Now <pz is to be associated with the parameter Xi of a new linkage; to remind us that this is a parameter of the inverted problem we shall distinguish it by a tilde: X\. Similarly (pi is to be associated with the parameter X2 in the new linkage. According to Eq. (69) we should begin the process of mechanizing the inverted function by choosing nb{2) = 0.265. To facilitate construction of the overlay we shall use an approximation to this: pb{*> = 0.25. (77) Such rounding off of values is generally useful in practical design work; we have here deliberately done it in such a way as to retard rather than accelerate the convergence of the method. We know that the linkage to be designed will not be very different, in its dimensions and in the arrangement of the scales, from that of Fig. 144 THE THREE-BAR LINKAGE [Sec. 5-14 ^7-0 5-17.- It must, however, differ from that linkage by reflection in a vertical line, since the pivots are to be interchanged; and it may differ also by reflection in a horizontal line. One can determine whether or not this additional reflection is involved by examining the <pi-scale; which, by the convention introduced in the discussion leading up to Eq. (56), must increase in the direction of increasing X2. In the present case it is evident that the two linkages must differ also by a reflection in a horizontal line ; the appearance of the new linkage will then be that of Fig. 5-17 turned upside down. One must accordingly expect X\m ~ —20°, and on the overlay will need to construct only a few curves of the plus family with s ~ — 2. Figure 5-18 shows the nomogram curve fib = 0.25 used in construction of the overlay, and the few lines of the plus family that need to be drawn. It is obvious that a good fit can not be obtained with curves for which rj is not a single- valued function of p. The curve s = — 2, for which we expect this fit, is, however, essentially single- valued. The retrograde portion of this curve closely overlaps the rest of it and is no bar to an accurate fit; its presence indi- cates only that 772 may reverse its direction of change as the linkage operates. When the overlay is turned about a horizontal line and moved over the nomogram a very accurate fit can be found between the overlay line 5 = — 2 and the line fxb2 = 0.275 of the nomogram, in the position indi- cated in Fig. 5-19. Also shown are the usual horizontal reference line and the auxiliary reference line \x-p = 0.3, which appears also in Fig. 5-18. From the position of fit we read the following values of the constants: 5 =-2 1 1 : 4 5 6 7 8 9 Fig. 5-18. — Scale and overlay for second application of the nomographic method. A2) A2) /z6(2) V(2) ^-2m V(2) Alm = 0.25, = 0.275, = 0.011, = 5° = -20°. as assumed, (78) Since the fit was obtained with a curve of the plus family, we have <P2 = Xi + 20°, 1 <Pi = X2 — 5°. J (79) Sec. 5141 AN EXAMPLE OF THE NOMOGRAPHIC METHOD 145 To make more evident the change in constants due to this second calculation, we can rewrite the above results in terms of the constants of the uninverted linkage. Remembering that the two linkages differ, in this case, by reflections in both horizontal and vertical lines, we have (80a) (806) Xi = X2 - 180°, Xlm = X2m ± 180° 9(f X2 = Xi - 180°, X2m-— X\m i 180 Fig. 5-19. — Position of fit in second application of the nomographic method. By use of these relations and of Eqs. (68), we have ixbf = -0.011, /x6(22> = 0.014, ^v = -0.25, Zffi = -175°, X™. = 160°, (81) 0.9975, A[2) = 1.778, B™ Af 1.837. These quantities represent distinct improvements over the first approximate values, except for jua(2) (which was rounded off in the wrong direction and not allowed to improve during the second fitting) and the ratios A2/Ai and B2/Ai (which depend upon fia2). In particular, the value of jib i deviates from the ideal by only one-tenth as much as the value initially assumed. It is evident that a second application of the nomo- graphic method to the mechanization of the given function in the direct form, with /x6i = —0.011, would lead to values of the constants very near to the ideal. THE GEOMETRIC METHOD FOR THREE-BAR LINKAGE DESIGN We shall now discuss a geometric method for the design of three-bar linkages for which the input travel AXi is not fixed but may be treated as 146 THE THREE-BAR LINKAGE [Sec. 5- 15 a variable parameter. This is a less common problem than that solved by the nomographic method, in which both input and output travels are treated as fixed; nevertheless, the method is a necessary and frequently useful complement to the nomographic method. The basic problem treated by the geometric method is that of finding the three-bar linkage with given values of AX2 and B2/A2 which most accurately generates a given function. In essence, the method is one by which a rapid comparison can be made between the desired and the actual positions of the input crank, for a series of positions of the output crank, for any given linkage of a large family. This comparison is made so easy that it becomes a relatively simple matter to find that linkage of the given family which gives the best fit. This solution can be improved, if desired, by a method of successive approximations like that employed with the nomographic method: the values of B2/Bi and AXi determined by the first application of the procedure are treated as fixed, and the initially chosen values of B2/A2 and AX2 improved by a second application of this procedure to the inverted function; then B2/Bi and AX\ are readjusted, and so on. When this method is employed, no constant of the linkage is held at an arbitrarily frozen value. 5-15. Statement of the Problem for the Geometric Method. — The problem to be solved by the geometric method is that of mechanizing a given functional relation, ft-ciz,)-*. h" i Xli XlMl (82) \_x2m ^ x2 ^ x2My as accurately as possible by a three-bar linkage with given output travel AX2 and given crank-link ratio B2/A2. A linkage will generate a relation X2 = (X,|X0 ' Xi (83) between its input and output parameters ; it will constitute a mechaniza- tion of the given function if there exists a linear relation between the parameters Xi, X2 and the variables xh x2\ Xx - X<°> = kfa - af>), (84a) X2 - Xf = k2(x2 - 40)). (846) Here X^ and x^ are corresponding values of X\ and xi, X(20) and xl20) corresponding values of X2 and x2 ; x[0) and xl20) do not stand in any neces- sary relation to the limits of the interval of definition in Eq. (82). In the problem at hand one knows both AX2 = X2M — X2m, (85) and Ax2 — x2M — x2m (86) Sec. 5- 16] SOLUTION OF A SIMPLIFIED PROBLEM 147 The magnitude of k 2 is thus determined: w = & (87) Also, it will be noted that a positive sign of k2 implies direct identification of the homogeneous parameters h2 and H2 corresponding to x2 and X2; a negative sign implies complementary identification. As in Sec. 5-9 we can, without loss of generality in the design process, assume direct identification of h2 and H2, while admitting either direct or comple- mentary identification of hi and Hi. Thus k2 may be considered as completely known, *. = -£> (88) Ax but ki is unknown both as to magnitude and sign. The fixed parameters of the problem are thus B2/A2 and k2; attention will be focused, in the actual design process, on the adjustment of Ai/A2, Bi/A2, and k\. 5*16. Solution of a Simplified Problem. — As in the case of the nomo- graphic method, we first consider a simplified problem in which there are only two adjustable parameters. Here we shall treat B2/A2j k2, and ki as fixed, and seek the best possible fit of the generated to the given func- tion by adjusting A i/A 2 and Bi/A 2. We reserve for Sec. 5- 17 an explana- tion of the method for varying k\. To solve this problem we choose a spectrum of values of the variable Xi\ r(0) r(l) ~(r) ~<n) extending through the interval of definition of Eq. (82). Equation (82) then defines a corresponding spectrum of values of x2\ ~(0) «(1) ~(r) ~(n) •*'2 ) *t/2 } • • • ) ^2 j • • • t "*'2 • Since both ki and k2 are known, Eq. (84) would define corresponding spectra of Xi and X2, if there were not present the unknown additive constants X[0) and X20). Given values of these constants, one could compute X[r) and X2r), and make sketches showing, for each r, the corre- sponding positions of the input and output cranks, each in its correct relation to its own zero position. Now, even though X^ and X20) are unknown, one can still compute such quantities as X[r+V - Xi" - ki(x[r+1) - x[r)), (89a) and X5TW>— XF = A^2(4r+1) - x2r)). (896) One can thus make a sketch showing the relative positions that the input crank must have for a sequence of values of r, and a sketch showing the 148 THE THREE-BAR LINKAGE [Sec. 516 corresponding relative positions that the output crank must have, if the given function is to be generated. Figure 5-20 shows such a set of relative positions for the input crank, represented by the radial lines B^ from the pivot point Si. The orientation of this figure with respect to the zero position of Xi — or, to put it another way, the direction on this figure of the line SiS2 between the crank pivots of the linkage — is unknown, since it depends on XJ*. Similarly, Fig. 5-21 represents, by the radial lines A2r) from the pivot point S2, the corresponding relative positions of the Xp*l>-XF Fig. 5-20. — The radial lines represent a series of relative positions of the input crank of a three-bar linkage. output crank; in this figure, too, the direction of the line SiS2 between the pivots of the linkage cannot be specified, since it depends on X(20). Figures 5-20 and 5-21 can be combined into a single figure representing a sequence of corresponding crank positions in the desired linkage, by placing them in proper relative positions. What the required relation- ship of these figures should be we do not yet know, but we do know enough about its characteristics to help us in finding it. For it is evident that (1) if the crank lengths A2 and B\ are laid out on the same scale, and (2) if the relative positions of the two figures are correct, and (3) if the given function can actually be generated by a linkage with the given B2/A2, kh and k2, then the distances between the ends of the cranks, in all corresponding positions, must be constant, and indeed equal to B2 Sec. 516] SOLUTION OF A SIMPLIFIED PROBLEM 149 on the chosen scale. By applying this idea one can determine the rela- tive positions of Figs. 5-20 and 5-21 which correspond to that three-bar linkage (with the given constants) which most nearly generates the given function; from the combined figure one can then read off the constants of this linkage. To understand how this can be done we consider Figs. 5-20 and 5-21 in more detail. The length A2 of the output crank has been taken as the unit of length in both Figs. 5-20 and 5-21. In Fig. 5-21, the points P(0\ Fig. 5-21. — The radial lines represent a series of relative positions of the output crank of a three-bar linkage ; the circles represent corresponding possible positions for the remote end of the connecting link. end of the connecting link P(r), . . . , P(n), represent a sequence of positions of the pivot T2 between the output crank and the connecting link. In Fig. 5-20, one cannot con- struct corresponding definite positions for the pivot T\ since the crank length Bi is unknown; instead, there is shown a sequence of circles of different radii, each of which defines, by its intersections with the radial lines, corresponding positions Q(0), . . . , Q(r), . . . , Q(n), of this pivot when the input crank has the appropriate length. In Fig. 5-21 there has been constructed about each point P(r) a circle C(r) having as its radius the known length B2 of the link; the remote end of the link, the pivot Th must lie somewhere on this circle. If it is possible to generate the given function by a three-bar linkage with the 150 THE THREE-BAR LINKAGES [Sec. 5-16 given constants, it must now be possible to place Fig. 5-21 on Fig. 5-20 in such a way that point Q.(0) lies on the circle C(0), point Qw on circle C(1), and so on, as shown in Fig. 5-22. The value of A i in the required linkage will then be the length of SiS2 on the common scale of the figures; the value of Bi will be the radius of the circle on which the points Qi0), . . . , Q(r), . . . , Q(n) lie; and successive configurations of the linkage will be defined by the points Sh S2, P(0), Q(0); Sh S2, P(1), Q(1); etc. Fig. 5-22. — Relative positions of Figs. 5-20 and 5-21 corresponding to a three-bar linkage generating the given function. In practical terms, the geometrical method of solving our restricted problem may be summarized thus: 1. Choose a spectrum of x\. 2. Compute the spectral values of x2, X\ — X^, X2 — X(20). 3. Construct Fig. 5-21 as a chart, on a sufficiently large scale. 4. Construct Fig. 5-20 as a transparent overlay, on the same scale. 5. Move this overlay freely, using both translation and rotation, over the chart, seeking a position such that the circles C(0), C(1), . . . , C(n) pass through the points Q(0), Q(1), . . . , Q(n), on some circle of the overlay. (In making this fit it may be necessary to consider each overlay circle in turn.) 6. If a fit is found, the unknown constants of the link can be read off, A i/A 2 as the distance SiS2, Bx/A2 as the distance SiQ(0), and Xim, X2m as the corresponding angles in the combined figure. Sec. 517] SOLUTION OF THE BASIC PROBLEM 151 7. If only an approximate fit is found, the error in the input angle, for any given value of the generated output angle, can be read as the angle subtended at Si by the arc from the corresponding Q point (for example, Qir)) to the intersection of the corresponding C circle (C(r)) with the arc Qi0)Q{nK Thus one should seek a position of the overlay which makes these errors as small as possible, and deter- mine the constants of the linkage as above. It will be evident to the reader that a change in sign of ki will leave Fig. 5-21 unchanged, but will produce the same effect on Fig. 5-20 as turning the overlay face down. A single overlay, used face up or face down, thus suffices for a given \ki\. 6*17. Solution of the Basic Problem. — We now turn to the basic problem of the geometric method, that of obtaining the best fit of the generated to the given function by simultaneous variations of three parameters of the linkage, keeping fixed the values of B2/A2 and k2. This can be accomplished without any essential complication of the procedure described in Sec. 5-16, by making a special choice of the spectrum of X\. This has also the advantage that the overlay corresponding to Fig. 5-20 then has the same form for all problems and can be used again and again. Let the spectrum of values x1? be chosen as x? = 40) + y^ '5, r - 0, 1, ". - • , n, (90) where 5 and g are constants such that all values x^ lie within the range of definition of Eq. (82). Equation (89a) then becomes xy-wi _ x? = M gr+1_~^r = kjg\ (91) The separations of consecutive spectral values X[r), the angles between successive positions of the input crank, will then change in geometrical progression. Figure 5-20 has, in fact, been drawn for such a case. So long as A: i is unknown, one cannot construct an overlay like Fig. 5-20. To overcome this difficulty we construct an overlay, Fig. 5-23, on which appear radial lines Lw with separations y«+D _ Ym = ag', t = 0, 1, 2, • • • . (92) (In principle, the sequence of t's might start with other values than 0; such cases can be reduced to the above by changing the choice of a and renumbering the fines.) Let us consider the n + 1 lines of this system labeled t = s, s -\- 1, • • • , s + r, • • • , s + n, with separations 7(a+r+l) _ y(«+r) = ag. . ^r (93) 152 THE THREE-BAR LINKAGE [Sec. 5-17 These will have the same separations as, and can be identified with, the lines B{°\ B?\ . . . , B{r\ . . . , B(n\ provided /bi5 = ags (94) or ag8 *i = (95) Thus by identifying various lines Z/s) of Fig. 5-23 as the line B^, one can in effect assign to ki any value given by Eq. (95) for an integral s. The ?3 24 25 Fig. 5-23. — Overlay for the geometric method. overlay is completed by the system of concentric circles which appears also in Fig. 5-20; it is used in the same way as that figure. The procedure is then as follows: (1) Choose a spectrum of values x({\ as given by Eq. (90). It is usually satisfactory to take g = 1.1; 5 may be positive or negative and should be so chosen that n, defined by - 1 9- 1 ^Asi, (96) Sec. 5-17] SOLUTION OF THE BASIC PROBLEM 153 lies in the range between 8 and 12. It is advantageous to choose the sign of 8, and the corresponding value of x^\ so as to make the spectrum of values x2r) as evenly spaced as possible. Thus in the case illustrated in Fig. 5-24a, in which dx2/dxi decreases as Xi increases, it is desirable to choose x^ at the lower end of the range of xi and to make 8 positive; when dx2/dxi increases as X\ increases, as in Fig. 5-246, z(0) should lie at the upper end of the range of xi and 8 should be negative. (2) Compute the corresponding spectral values cf x2 and X2 — X2°\ using Eqs. (82) and (846). JO) *2 4°> rW) *£> *1 I 6>0 6<Q (a) (6) Fig. 5-24. — Choice of xi(o) and 5 to make the spectral values x% (r) as evenly spaced as possible. (3) Construct a transparent overlay similar to Fig. 5-23, with succes- sive radial lines at angles yw = gl (97) measured clockwise from the zero line. The value of g must be the same as that chosen in Step (1); a may be chosen arbitrarily but should be small. Figure 5-23 has been drawn with g = 1.1, a =* 1°. Label each radial line with the corresponding value of t. (4) Using the spectral values of X2 — X20), construct a chart corre- sponding to Fig. 5-21. The length of the crank, A2, should be one unit on the scale used in constructing the overlay. Lay down the successive crank positions, A2r), and about the end points P(r) construct circles C(r) with the known radius B2. (5) Place the overlay on this chart, face up, and seek a position for it such that the (n + 1) circles, C(0), C(1), C(n), on the chart pass through (n + 1) points Q(s), Q{s+1), • • • Q(sfr° on the overlay in which a circle of the concentric family, labeled J?i, intersects n + 1 consecutive radial lines, Z/8), Z/s+1>, . . . U8+n\ 154 THE THREE-BAR LINKAGE [Sec. 5-18 In seeking this fit one has to consider: a. All possible positions on the chart of the point S\ of the overlay. b. All orientations of the overlay — i.e., all values of s. c. All circles of the concentric family. The problem is not as difficult as it might seem. Let the point Si of the overlay be placed in a fixed position on the chart. Each circle of the overlay will be intersected by the C circles in a sequence of points which will be unchanged by rotation of the overlay. Unless successive intervals between these points change in geometric progression, by a factor g, there is no possibility of obtaining a fit by turning the overlay. Thus, for each position of the overlay center Si, a quick inspection of the spacings of the intersections of the two families of circles will suffice to determine whether there is any chance of a fit on any circle of the overlay. By a systematic survey of this type one can reject large areas of the chart as possible positions for Si. When a sequence of intersections has been found in which the intervals change in about the right way, it becomes worth while to turn the overlay until the radial lines in the region of intersection have similar spacings — for example, until an s is found such that circles C(0) and C(n) pass through the points Qw and Q(s+n), respectively. This configuration will corre- spond to a linkage in which the errors in the generated function would vanish at the ends of the range of xi; the errors in the generated function in the intermediate range are evident, being measured by the angular distances on the overlay between the points Q(s+r) and the intersections of the circles C(r) with the Bi overlay circle. With practice one rapidly develops a technique for improving this fit by smaller adjustments in the position of Si, with corresponding rotations of the overlay. (6) If an acceptable fit is not found with the overlay face up, turn the overlay face down, and repeat the process. (7) When a fit has been found, the elements of the linkage can be read directly on the overlay scale; B1/A2 is the value of B for the overlay circle on which the fit is obtained, and A1/A2 is the value of B for the overlay circle that passes through the point S2 on the chart. Limiting configurations of the linkage are evident from the arrangement, and values of Xim, X2m, and Xm can be read. Figure 5-22 actually represents an application of this method, since Fig. 5-21 is, in fact, the portion of Fig. 5-23 in which s changes from 22 to 30. A full example of the method is presented in Sec. 5-19. 5-18. Improvement of the Solution by Successive Approximations. — A first solution of the problem of mechanizing a given function can be improved by successive applications of the geometric method, in essen- tially the same way as with the nomographic method. Sec. 518] SOLUTION BY SUCCESSIVE APPROXIMATIONS 155 The first approximate solution will have been found with fixed values of the constants AX2 and B2/A2. The first of these constants may be determined by other factors in the problem, but the choice of the second will have been to some degree an arbitrary one. If the choice of B2/A2 was very unfortunate, the fit obtained may be so bad that the process must be repeated with another value of this constant. In most cases one will find a reasonably good mechanization- of the function — one which is at least sufficiently good to serve as a guide in finding a better one. In particular, note should be taken of the values found for the constants B2/Bi and AXi of this linkage. Now let us consider the inverse of the function of Eq. (82), Xi = (xi\x2) ' x2, (98) with x2 treated as the input variable. Interchanging the roles of xi and x2 in Sees. 5-16 and 5-17, one can apply the geometric method to the mechanization of this relation and thus obtain a second mechanization of the original relation. The inverted problem differs from the original in the interchange of B1 and A2, Xi and 180° - X2 (cf. Sec. 543). Thus it is evident that appropriate choices for the fixed constants of the new problems are Bp ^Bp ) Ii» Bp \ (99) If the conditions of the problem dictate a special choice of AX2, one should treat AX12) = AXp also as a constant; the problem is then that discussed in Sec. 5-16. [It can, of course, be treated by the method of Sec. 5-17, with s restricted to a constant value determined by Eq. (94) or Eq. (95)]. In other cases one will treat AXi as a variable parameter in the inverted problem. In any case the inverted function will be approximated by a linkage selected from a family which includes the mirror image of the original linkage ; the fit, if properly made, must be at least as good as that found as a first approximation, and will usually be appreciably better. A third approximation can then be found by returning to the considera- tion of the uninverted function and applying the geometric method with the fixed constants Bg> ^ B? ] Af gp' I (100) AX(23) = AX<2). I As a rule this process converges toward a certain optimum solution of the problem. It is to be noted, however, that there may be several such 156 THE THREE-BAR LINKAGE [Sec. 5-19 approximate solutions within the class of three-bar linkages; which of these is found will depend upon the initial choice of B2/A2 and AX2. When one finds a mechanically unsatisfactory solution of the problem, it is usually profitable to start the process again with a different value of B2/A2. In applying the geometric method it will be found that the values of AXi and AX2 converge more rapidly to a limit than do the ratios of the sides of the quadrilateral. It is therefore suggested that this method be abandoned as soon as the values of AXi and AX2 are sufficiently well determined, the calculation being completed by the nomographic method. 6-19. An Application of the Geometric Method : Mechanization of the Logarithmic Function. — We shall now apply the geometric method to the mechanization of the logarithmic function x2 = logio xi (101) in the range 1 < xi < 10, 0 < x2 < 1. (102) In terms of the homogeneous variables fc, = ^^ (103) h2 = x2, (104) the relation to be mechanized becomes ht = log10 (9/i! + 1). (105) Since the logarithmic function is of the type illustrated in Fig. 5 -24a, we shall choose a positive 5. The spectrum of values of the homogeneous variable hi can then be written as hf = 0, (106) h[r) = 9r 9 ■ — 1 1 K Mn) 9n - 1 • 5. 9- 1 We shall choose g = 1.1, n = 10. Solution of the last of Eqs. (106), with /i(!n) = 1, gives d = 0.0627. (107) The values of h<p can then be computed by Eq. (106), and the correspond- ing values of h(2r) by Eq. (105). The resulting values are shown in Table 5-2. Sec. 519] MECHANIZATION OF THE LOGARITHMIC FUNCTION Table 5-2. — Spectral Values for the Logarithmic Relation 157 r h[r) /4r) 0 0.0000 0.0000 1 0.0627 0.1943 2 0.1318 0.3397 3 0.2077 0.4578 4 0.2912 0.5588 5 0.3830 0.6481 6 0.4841 0.7289 7 0 . 5952 0.8032 8 0.7175 0.8726 9 0.8520 0.9379 10 1.0000 1.0000 If we express this relation in the inverted form, treating x2 or h2 as the input variable, the function is of the type shown in Fig. 5-246. In mechanizing this by the geometric method the spectral values of h2 should be chosen with 8 negative. Distinguishing by a tilde the spectral values required in this inverse mechanization, we have M0) = 1 Mr> = 1 + JL (108) 1 + - 1 8 = 0. With g = 1.1, n = 10, as above, one finds, on solving the last of these equations, the same magnitude as before for 8 : Thus 5 = -0.0627. h%> = 1 - h[r); (109) (110) the corresponding values of h({\ computed by Eq. 105), are shown in Table 5-3. We begin mechanization of the relation in the direct form by choosing arbitrarily Bp ^ = 1.25; AXp = 100c (111) The overlay required for the work is determined as soon as g and an arbitrary small angle a are chosen; with a = 1° it has the form shown in Fig. 5-23. The chart to be constructed depends, however, on the 158 THE THREE-BAR LINKAGE [Sec. 5-19 Sec. 519] MECHANIZATION OF THE LOGARITHMIC FUNCTION 159 Table 5-3. — Spectral Values for the Logarithmic Relation in Inverse Form r Up h[r) 55° X h[r\ degrees 0 1.0000 1.0000 55.0 1 0.9373 0.8506 46.8 2 0.8682 0.7092 39.0 3 0.7923 "0.5777 31.8 4 0.7088 0.4572 25.1 5 0.6170 0.3489 19.2 6 0.5159 0.2533 13.9 7 0.4048 0.1711 9.4 8 0.2825 0.1018 5.6 9 0 . 1480 0.0451 2.5 10 0.0000 0.0000 0.0 particular problem here considered. On this chart (cf. Fig. 5-25) the lines A(2r) radiate from the point S2, making angles h2r)AX2 = h2r)100° with the zero line. The points P(r) lie on these lines at unit distance from >S2. About each of these is drawn a circle C(r) with radius J = B, = 1.25. This completes preparation of the equipment. The overlay is now placed face up on the chart, and it is found (as shown in Fig. 5-25) to be possible to make the circles C(r) pass, approximately, through the points Q(o) . . . quo) at wmch the interpolated circle B = 0.95 on the overlay (dashed circle in Fig. 5-25) intersects the radial lines L(21) to L(31). The tit, however, is rather poor at the points Q(1) Q(8), Q(9). In addition, the linkage would be mechanically unsatisfactory because of the small angles between the output crank and the link at small r, and between the input crank and the link at large r. (The extreme configurations are indicated by dashed and heavy solid lines in Fig. 5-25.) The fit could be improved by the method of Sec. 5-18, but the approximate solution thus found would probably have the same unsatisfactory mechanical characteristics. No satisfactory fit can be obtained by turning the overlay face down. We therefore repeat the process with another choice of B2/A2. We now try ffl -18 AXj," = 100c (112) The overlay is unchanged, and the chart is changed only in that the circles C(r) have the larger radius B2 = 1.8. The same chart can thus be used again, with the new circles drawn in ink of another color. A more satisfactory fit can now be obtained, this time with the overlay face 160 THE THREE-BAR LINKAGE [Sec. 5-19 Sec. 5-19] MECHANIZATION OF THE LOGARITHMIC FUNCTION 161 down (Fig. 5-26); the circles C(0) to C(10) pass very nearly through the points Q(0) to Q(10), at which the overlay circle B = 1.2 intersects the radial lines L(13) to L(23). From this figure one reads the constants of the linkage: Hence (1) O) = 1.8; £i- = 12- ^-15 AP AV = 1.79. (113) (114) The angle AXi can be measured on the overlay, but is even more easily obtained as the difference of tabulated values of Yw : AXi = Y*"** - F<s) (115) These values are given, for the overlay Fig. 5-23, in Table 5-4. In the present case AXi - F<23> - F<13> = 89.54° - 34.52° = 55.02°. (116) Table 5-4.— Y«\ for g = 1.1, a = 1° * Y(t), t Y(t), t Y(t), t Y(t), degrees degrees degrees degrees 0 10.00 10 25.94 20 67.27 30 174.49 1 11.00 11 28.53 21 74.00 31 191.94 2 12.10 12 31.38 22 81.40 32 211.14 3 13.31 13 34.52 23 89.54 33 232.25 4 14.64 14 37.97 24 98.50 34 255.48 5 16.11 15 41.77 25 108 . 35 35 281.02 6 17.72 16 45.96 26 119.18 36 309.13 7 19.49 17 50.54 27 131.10 37 340.04 8 21.44 18 55.60 28 144.21 38 374.04 9 23.58 19 61.16 29 158.63 Although the linkage thus obtained is not mechanically satisfactory when r is small (xi and x2 near their lower limits), we attempt to improve it by application of the geometric method to the inverted function, with B<P = £i- = 1.5 and AX(22) = AX^ = 55c (117a) (1176) A completely new chart must be constructed, with radial lines A%"> making angles h[r)AX2 = Mr)-55° w^h the zero line; the required values will be found inJTable 5-3. (It must be remembered that in this inverse problem hi and X2 vary together, as do h3 and Xi. In the procedure, ~h[r) now 162 THE THREE-BAR LINKAGE [Sec. 519 Sec. 519] MECHANIZATION OF THE LOGARITHMIC FUNCTION 163 takes the place of /i2r) ; the values of ~h{? have been so chosen that the differences increase in geometrical progression and correspond to succes- sive lines on the overlay.) The points P(r) are constructed at radius A^ = 1, and about these are drawn circles C(r) with radius S(22) = 1.5 (Fig. 5-27). When the overlay is placed on this chart, face up, the circles C(r) can be made to pass very nearly through the points Q(r) at which the circle B = 0.75 intersects the radial lines Z/18) to L(28); on a larger scale it can be seen that the fit is perhaps a little better than that obtained in the preceding step, but the accuracy obtained in both cases is about the best that can be expected of the geometric method. One reads from the figure = 1.39; (118a) (1186) AZ(2) = F(28) _ F(i8) = 88.61°. (119) In terms of the constants of the uninverted problem the above results become Z?(2) A (2) A (2) £l_ - 1 * til- - 0 7^ l - 1 3Q #1 Bi #i (120) R(2) ^- = 1.5, A? 52 - 0.75, A? If hence R(2) £- = 2.0. The input angular range is and /?(2) ^ = 2.0. AX(22) = 88.61°. (121) The values of B^/Af* and AX(22) are not very different from those of Eq. (112), with which we started; it is evident that the solution is not far from the best one — or, at least, the best one with approximately these constants. It is therefore reasonable to fix on definite travels, AXi = 55°, AX2 = 90°, (122) as sufficiently close to the best values, and to determine a final design using the nomographic method. The reader will find it a useful exercise to carry through this step, using the procedure of Sec. 5-11. We have, by Eq. (51), <Pi-hi- 55°, (123) <P2 = h2 • 90°. 164 THE THREE-BAR LINKAGE [Sec. 5- 19 Table 5-5. — Spectral Values of the Parameters „/r> Jrt r degrees *P ti? degrees 0 0 0.0000 0.0000 0.0 1 10 0.1818 0.4209 37.9 2 20 0.3636 0.6306 56.8 3 30 0.5454 0.7715 69.4 4 40 0.7272 0.8777 79.0 5 50 0.9090 0.9627 86.7 6 60 1.0908 1.0341 93.1 To make it possible to use Fig. Bl, we choose 8 = 10°, though, in view of the small value of AXh it would be better to use 5 = 5°. The spectral 10*1 Fig. 5-28. — Approximate mechanization of x-i = logio Xi. values ^ir) and ^(2r), computed with the aid of Eq. 105, appear in Table 5-5. The choice of fxbi suggested by the last application of the geometric method [Eq. (120)] is ubi = logi log,0 (1.39) 0.143 « -0.15. (124) Sec. 519] MECHANIZATION OF THE LOGARITHMIC FUNCTION 165 Only a few lines need be drawn on the overlay. Picturing the mirrored form of Fig. 5-27 with Si to the left of S2, one sees that the scales of <& and X2 increase together, whereas <p% and Xi increase in opposite senses; the fit is to be expected with an overlay curve of the minus family, probably with s = 15, since X\M ~ 150°. Choosing M&i = -0.15, we find Hence nb2 = 0.283, fia = 0.260, Xim = s8 = 150°, X2m = -116.5°. *i = 0.707, ^ = 1.919, 41 = 1-820, Ax A2 A2 (125) and finally } (126) ^1 = 1.055, 4^ = 0.550. Ax Ai The linkage is sketched in Fig. 5-28. It will be discussed further in a later example (Sec. 7-8). CHAPTER 6 LINKAGE COMBINATIONS WITH ONE DEGREE OF FREEDOM It is only rarely that one can mechanize a given function with high accuracy by a harmonic transformer or a three-bar linkage. Usually a more complex linkage must be employed in order to gain the flexibility required in fitting the given function with sufficient accuracy. Instead of devising entirely new structures it is better to combine the elementary linkages; the double harmonic transformer discussed in Chap. 4 is such a combination. Other useful combinations are the double three-bar linkage — analogous in structure to the double harmonic transformer — and combinations of single or double three-bar linkages with one or two harmonic transformers. Choice of the proper combination should of course be determined by the type of function presented for mechanization. Techniques for the design of such linkages will be indicated in the present chapter. COMBINATION OF TWO HARMONIC TRANSFORMERS WITH A THREE-BAR LINKAGE 6*1. Statement of the Problem. — The combination of two harmonic transformers with a three-bar linkage, as sketched in Figs. 6-1 and 6-2, is particularly useful when it is desirable to use slide terminals at both input and output. (In these figures both harmonic transformers are indicated as ideal; in practice both will usually be constructed as nonideal.) The input link and the crank RiSi constitute a harmonic transformer that transforms the homogeneous input parameter Hi into the homogeneous angular parameter 0i. The angular parameter corresponding to 0i will be called X\ (Fig. 6-2); the constants of the harmonic transformer are then Xim, AXi. (It is important to remember that 0i, not Hh is the homogeneous parameter corresponding to X%.) The crank TySi, rigidly linked to RiSi, is described by an angular parameter X3 and a homo- geneous angular parameter 03, which will be identically equal to 0i. The input harmonic transformer thus carries out the transformation: 03 = (03|#i) ■ Hi. (1) The cranks Ty3\ and T2S2, with the link T1T2, form a three-bar linkage (constants X3m, AX3 = AXh X4m, AX4, etc.) that transforms the param- eter 63 into another homogeneous angular parameter, 04 = (OM ■ 03, (2) 166 Sec. 6-1] STATEMENT OF THE PROBLEM 167 associated with the angular parameter X4. The crank R2S2} rigidly linked to T2S2, is described by the angular parameter X2, or by the homogenous angular parameter 02, identically equal to 04. Finally, the crank R2S2 and the output link form a harmonic transformer (constants X2m, AX2 = AX4), which trans- ^ forms 04 = 02 into the homogeneous output parameter H2 = (#2|04) • 04. (3) Fig. 6-1. — Three-bar linkage combined with two harmonic transformers. It will be noted that the angles Xi and X2 describing the harmonic transformers cannot in general be measured from the same zero lines as the angles X3 and X4 describing the three-bar-linkage configuration, if the conventions of the preceding chapters are to be maintained. In the particular cases illustrated in Figs. 6-1 and 6-2, in which the input and output links of the transformers are parallel to the line of pivots of the three-bar linkage, the zero lines for Xi and X2 are perpendicular to those for X3 and X4. >4. ^ Fig. 6-2. — Combination of three-bar linkage with two harmonic transformers, sketched in its extreme positions. The linkage as a whole carries out the transformation H2 = (H2\H{) • Hi, where (H2\Hi) = (H2\94) • (04|03) • (03|#i). Given a functional relation in homogeneous form, h2 = (h2\hi) - hh (4) (5) (6) one will wish to find harmonic-transformer functions, (#2|04) and (03|#i), and a three-bar-linkage function, (04|03), such that the product operator (H2\Hi) will approximate as closely as possible to (h2\hi), on direct or 168 LINKAGE COMBINATIONS [Sec. 6-2 complementary identification of the parameters (Hh H2) with the variables (hi, h2). It would be very difficult to find the best approximation to {h2\hi) within the twelve-parameter family of available functions. The tech- nique to be described is intended only as a practically useful method for obtaining a good result in a reasonably short time. This involves a preliminary resolution of the desired operator (H2\Hi) into three factors: two harmonic transformer operators (usually ideal), and a third operator to be mechanized by the three-bar linkage. When the three-bar linkage has been designed, by the methods of Chap. 5, the harmonic transformers are redesigned, almost invariably as nonideal, in order to get a better fit to the given function. Finally, the over-all error is further reduced by small simultaneous variations of all constants of the linkage, by methods to be discussed in Chap. 7. 6-2. Factorization of the Given Function. — A rapid method for finding a satisfactory preliminary factorization of (H2\Hi) is essential to the success of this procedure. Let Eq. (5) be multiplied from the left by (6i\H2), from the right by (#i|03). One obtains (04|03) = (d4\H2) • (H2\Hi) • (Hi\6s). (7) Of the quantities on the right, (H2\Hi) has a prescribed form in the given problem, and the operators (6A\H2) and (i/i|03), though unknown, are of a relatively limited class — particularly when attention is restricted to the ideal-harmonic-transformer operators of Tables A-l and A-2 in carrying out the preliminary factorization. More or less reasonable choices of the operators (04|#2) and C£/"i|03) can be based on consideration of the form of the given function. One can then quickly determine, by the graphical multiplication corresponding to Eq. (7), the required form of (04|03). Inspection of this function will suffice to indicate whether it can be approximated by a three-bar-linkage function. If so, the constants of that linkage can be found by the methods of Chap. 5; if not, the problem must be reconsidered and another choice of harmonic-transformer func- tions tried. This process of trial and error is not excessively burdensome since each trial involves only reference to Tables A-l and A-2 and a graphical construction. The speed with which it can be carried out depends, of course, on the judgment and experience of the designer, both in selecting the harmonic-transformer functions and in assessing the possibility of mechanizing the derived (04|03) by a three-bar linkage. Some suggestions on the first of these matters are contained in the follow- ing paragraphs. It is possible, though not usually desirable, to mechanize a given function approximately by a double harmonic transformer and to use an interposed three-bar linkage to make a small correction; it will rarely be Sec. 6-2] FACTORIZATION OF A FUNCTION 169 satisfactory to mechanize the given function approximately by a three-bar linkage and then attempt to convert to slide input and output by harmonic transformers that make only small changes in the form of the generated function. Instead, in mechanizing monotonic functions it is better to make all three components of the linkage combination contribute about equally to the curvature of the generated function. Perhaps the simplest way of accomplishing this is to focus attention on the terminal slopes of the factor functions, which become more and more different from 1 as the curvature increases. When all factor functions are monotonic one has (*Ei) =(^1*) .-(&*) .(ih) , (8a) \dHi/Hi=0 \dd4 /04=O \G?03/03=O \dHi/Hx=0 (dH\ =(dlh\ (d(U\ (ddA . Wi Ax=i \ de, Je^i \deje3=i \dHjBvmlf ^0) the terminal slopes of the generated function are products of the corre- sponding terminal slopes of the factor functions. For a first orientation, to make sure that no factor function need have excessive curvature, one can require that all factor functions have the same terminal slopes: (d]h\ /dtfA ^(dds\ ~[(*H*\ Y (M \dej0i=o \ddJe3=o \dHjHl=o LWiAi-oJ ' V ; Specification of both terminal slopes is sufficient to fix the ideal-harmonic- transformer functions completely; they may be identified by reference to Figs. 4-17 and 4-18. By use of Eq. (7) one can then determine the corresponding required form of (^4^3), for examination as to the possi- bility of mechanizing it by a three-bar linkage. It is to be remembered that this linkage must be one of specified angular travels AX3 and AX4, these being fixed as the angular travels of the input and output harmonic transformers, respectively. If there exists no ideal-harmonic-transformer function with the speci- fied terminal slopes, or if the angular travels AX! = AX3 and AX2 = AX4 are unsatisfactory, one can lighten the restrictions on the terminal slopes by requiring only that /tf#A (deA \ ddi / 04=0 \dH 1/ Hl=o (dH3\ /dOt\ \(dH*\ \ dd* /*4=1 \dHjHl-i l\dHjBlmml ( dH2\ dH\t Bl=o (10a) (10b) In addition, any convenient angular travels AXi and AX2 can be specified and the constants of the two ideal harmonic transformers then deter- 170 LINKAGE COMBINATIONS [Sec. 6-2 mined by use of Figs. 4-17 and 4-18. (An example is provided in Sec. 6-3.) The required three-bar-linkage function, found as before, will again have terminal slopes given by Eqs. (9). When the given function has one maximum or minimum, at least one of the three factor functions must also have a maximum or minimum. Only one of Eqs. (8a) and (86) can then be valid, and a different pro- cedure must be employed. It is usually best to choose the output- harmonic-transformer function as nonmonotonic — that is, to attempt to mechanize the function by a linkage of the sort illustrated in Fig. 6-2. The constants of this transformer should be such that the function to be generated by the other two elements of the combination, (04|#O = (04|08) * Mffi) = (0A\H2) ■ (ff2|#i), (11) is monotonic and as smoothly curved as possible. The function (6A\Hi) will be monotonic only if the harmonic-trans- former function (i72|04) has the same values as the given function (jEf2|-ffi) at the ends of the range of vari- ables (Fig. 6-3). This require- ment fixes the form of (#2|04), and hence (64\H{), for any given AX2; it remains to choose the value of this constant. This should be done with some attention to mechanical suitability but pri- marily so as to assure that (04|#i) is a smoothly curved function, as in Fig. 6-3; it is more important to avoid points of inflection in (04 1 #1) than to make its curvature small. When (H2\0t) has been determined and the corresponding function (04|#i) has been obtained by graphical construction (Sec. 3-4), it remains to resolve this latter function into the product on one harmonic-transformer function and a three-bar-linkage function, as expressed in the first part of Eq. (11). As when resolving a given function into three factors, it may here be desirable to choose the harmonic-transformer factor by reference to its terminal slopes, fixing (H2\H1) -> V s 7 ^ ^ r y / / / / \Ht *l«4> v/ / * / / / "-— ( e4ii y / Fig. 6-3. — Resolution of a given function (Hz\H\) into an output-harmonic-trans- former function (H2\di) and a monotonic function (04|#i). (dd3\ (d0*\ \dHjHl=l (de*\ \dHj„1=o (ddA (12a) (126) Sec. 6-3] EXAMPLE: FACTORING THE GIVEN FUNCTION 171 This will determine the constants of the second harmonic transformer, and it will remain only to work out the required form of the three-bar- linkage function by a second graphical construction. In some cases it will not be possible to satisfy both of these conditions; one can then, for instance, satisfy one or the other, and in addition fix AXi. It is to be emphasized that the preceding paragraphs do not contain a prescription that assures immediate success, and are intended only to be suggestive. A satisfactory resolution of the function may be found only after several trials, in which the designer must be guided by his imagina- tion and experience. Sections 6-3 and 6-4 will carry an example through the stages of factorization of the given function and mechanization of the three-bar- linkage factor, to the point where there is obtained a first approximate mechanization of the given function by a combination of a three-bar linkage and two ideal harmonic transformers. In Sec. 6-5 we shall then return to a general discussion of the next stage of the design procedure — improvement of the fit by introduction of nonideal harmonic transformers. 6*3. Example: Factoring the Given Function. — To illustrate the details of the method we shall consider again the problem of mechanizing the tangent function, but through a wider range of variables than was attempted in Chap. 4: x2 = tan xi, 0 < xi < 80°. (13) As usual, we introduce homogeneous variables, X\ hi = gQO> (14a) ^ = JMTz ^ Table 6-1. — x2 = tan xh 0 ^ x± ^ 80°, in Homogeneous Variables hi h2 0.0 0.0000 0.1 0.0248 0.2 0.0506 0.3 0.0785 0.4 0.1102 0.5 0.1480 0.6 0.1958 0.7 0.2614 0.8 0.3615 0.9 0.5427 0 . 95 0 . 7072 1.0 1.0000 172 LINKAGE COMBINATIONS [Sec. 6-3 Equation (13) then becomes ht = 0.17632 tan (hi ■ 80°) = (h*\hi) • hi. (15) This function is tabulated in Table 61 and plotted, as the desired (H2\Hi), in Fig. 6-4. The terminal slopes, obtained by differentiating Eq. (15), are (i;L=°-246> (£L=8-165- <»> We first consider the possibility of applying Eqs. (9) in factoring this monotonic function. This would require (SfL = (SL = (0246)M = °-627> (17a) (SfL = OS)*- = (8165P = 2013- (m) Inspection of Figs. 4-17 and 4-18 shows that there exist no ideal-harmonic- transformer functions with the required terminal slopes; it is necessary to use the lighter conditions of Eqs. (10), which become /dHA \ rf#4 / 04=Q _ \dds /e,=o \dd4 /04=i _ (dHi\ \ d6a Je^i (0.627) 2 = 0.393, (18a) (2.013)2 = 4.057. (186) In addition, values can be assigned to AXi and AX2. If one requires that AXi = AX2, the problem becomes identical with that discussed in Sec. 4-11. Applying the method of solution described there, one finds, for example, the following sets of constants that satisfy Eqs. (18) : 1. AZi = AX2 = 90°, Xlm = -7.5°, X2m = -67.5°. 2. AX i - AX2 = 100°, Xlm = -17.5°, X2m = -70°. Other sets of constants with AXi = AX2 are easily found by the same method; a slight and obvious modification of the method is required if one wishes to have AX\ ^ AX2. For a first trial we shall choose AXi = AX2 = 100°, these values being both mechanically satisfactory and especially convenient for the computations to be made. Then the harmonic-transformer functions to be used are Sec. 6-3] EXAMPLE: FACTORING THE GIVEN FUNCTION 173 (Hl\ds): -17.5° ^ Xi ^ 82.5°, (Ht\eA): -70° ^ X2 ^ 30°. These functions are plotted in Fig. 64, the first as a set of encircled points, the second as a continuous curve. The desired form of the three-bar-linkage function can now be com- puted by application of Eq. (7), or some equivalent equation. As is 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 6-4. — Resolution of the tangent function, (Hi\Hi), into a product of harmonic- transformer functions, (#2104) and (03|Hi), and a function (04|03) that is to be mechanized by a three-bar linkage. usually desirable, in Fig. 6-4 we have plotted all functions with the 0-scale horizontal. (Systematic use of this convention helps to prevent mistakes and makes easier the change to nonideal transformers.) To make use of these plots in a graphical construction for (6 ^63) we must, in effect, solve an equation that involves harmonic transformer functions only in the form (H\0). This we can obtain by multiplying Eq. (7) from the left by (H2\BA): (H2\e4) • (04|03) = (#2|#i) • (H1\e3). (19) 174 LINKAGE COMBINATIONS [Sec. 64 The product on the right can be formed by graphical multiplication of known operators, as in Fig. 64, where construction of the rectangle 0 — > A — > B — > C leads to location of the point C on the (unplotted) curve representing this product. A corresponding point F on the curve of (04|03) is then found by graphical solution of Eq. (19), through construc- tion of the rectangle C — * D — > E —+F, It is, of course, unnecessary to locate point C. The complete construction for the point F of the function (04|03), corresponding to the point 0 of the function (i/i|03), then involves (1) construction of a vertical line through the point 0, and (2) location of the point F by construction of the lines 0 — * A —*B^>D—>E—*F. The complete curve (04|03) shown in Fig. 6-4 is quickly determined by repeated application of this construction. It appears to be a function that can be mechanized by a three-bar linkage. We therefore tentatively accept the resolution of the given function as satisfactory, and turn, in Sec. 6-4, to the problem of designing the corresponding three-bar-linkage component. 6-4. Example: Design of the Three -bar -linkage Component. — We have now to consider the problem of mechanizing a function, given graphically in Fig. 6-4, by a three-bar linkage with fixed angular travels, AX3 = AX4 = 100°. (20) Since both angular travels are fixed, the nomographic method must be used in determining the other constants of the linkage. Although this method has been illustrated in Sec. 5-14, it may be desirable to show all stages of the procedure also in the present case, which differs from the earlier example in that the method of successive approximations described in Sec. 5-13 converges very slowly. This example will also serve to illustrate the fact that a given function can often be mechanized by several quite different linkages, among which one must make a choice on the basis of mechanical suitability. The function to be generated by the three-bar linkage, as read from a carefully constructed chart, is given in Table 6-2 in both the direct and the inverted form. The variables used are not the homogeneous variables 03 and 04, but the angular variables [cf. Eq. (5-51)], <pi = AX303, (21a) <p2 = AX404. (216) (It is to be remembered that in this example X3 will replace the Xi of Sees. 5-7 to 5-13, and X4 will replace X2.) We begin by taking p6f) = _o.2, (22) the usual first choice of the author. Sec. 6-4] EXAMPLE: DESIGN OF A THREE-BAR LINKAGE Table 6-2. — Given Function for the Example 175 falrO (<pi 1*0 <pi, <Ph ¥>h <Pt, degrees degrees degrees degrees 0.0 0.0 0.0 0.0 6.7 10.0 15.5 10.0 12.6 20.0 32.4 20.0 18.5 30.0 47.3 30.0 24.7 40.0 59.2 40.0 32.1 50.0 68.9 50.0 40.7 60.0 76.4 60.0 51.3 70.0 82.8 70.0 65.5 80.0 88.6 80.0 82.5 90.0 94.3 90.0 100.0 100.0 100.0 100.0 Choosing 5 = 10°, n = 100°/10° = 10, we find the values of ^ and <p[r) in Columns 1 and 2 of Table 6-2. Construction of the overlay follows precisely the steps described in Sec. 5-14 and need not be explained here. On turning the overlay face down on the nomographic chart, a satisfactory fit is found between the curve s = —8 of the minus family on the overlay, and the curve fib 2 = 0.075 supplied by interpolation on the nomogram. Figure 6-5 shows on the nomogram grid the con- struction of the particular overlay curve for which the fit was obtained, and the position of fit on the chart. The fit is exact at the ends, very good for the larger values of up and the smaller values of X3, and some- what less satisfactory for the larger values of X3. The reference lines of the turned overlay are also shown in the position of fit. The elements of the linkage are thus established: iu.a fib^} = —0.2, as assumed. ttb? = 0.075. = 0.207, read at the intersection of the vertical reference line with the jup-scale. X& = — 7.2°, read at the intersection of the horizontal reference line with the 77-scale. = —80°. (sd = Xzm, not Xsm, because the curve that gives a fit is of the minus family.) xffi Xffi = X',1' - AX',1' = -180°, 3M 176 LINKAGE COMBINATIONS [Sec. 6-4 By Eq. (5-56) we have (using the lower signs in the first equation because the fit was obtained with a curve of the minus family) (pi = —X3 — 8 <p2 = X4 + 7.2C (23a) (236) -60 -90° Fig. 6-5. — Mechanization of the tangent function: First application of the nomographic method. The dashed line is the interpolated contour fib = 0.075. Equations (23) are only as exact as the fit obtained. Since the fit is exact at the ends of the range of X3, the input and output travels of the linkage as designed will have the required value, 100°. The <pi- and 03-scales will increase with decreasing X3, and <p2, 04, and X4 will increase together. As a check, the linkage is drawn as in Fig. 6-6, which shows the cranks in their extreme positions. The distance A^ between the crank pivots is taken as the unit of length; the relative crank lengths are drawn in as A{» A? A[l) = 10"*1 = 10-°-2 = 0.630, = io-m« = io~c = 0.620. (24a) (246) Sec. 6-4] EXAMPLE: DESIGN OF A THREE-BAR LINKAGE The constancy of the required length of the connecting link, (Bp\ (Al\ _ \APj\AiV " = IQoibr-a) = 10-0.132 = Q 73^ 177 (25) provided a check on the quantities determined in the fitting process. The fit obtained with this first value of ubi is so good that one cannot expect it to be changed greatly by further calculations. Nevertheless, we now attempt to improve it by the process of successive approximations, interchanging the roles of <p\ and cp2. The inverted function (Sec. 5-13) has already been given in Columns 3 and 4 of Table 6-2. We begin the process of mechanizing this by taking M5j2) = -0.2 « -mg(1), the approximation being close enough for the purpose at hand. (26) Fig. 6-6. — Mechanization of the tangent function: First three-bar-linkage design. We know that the linkage to be designed will not be very different from that of Fig. 6-6 reflected in a vertical line. For the inverted prob- lem we must then have X1M « - 170°, s « - 17. (The fit will again be found in the minus family of curves.) Thus only a few lines need be drawn on the overlay. Figure 6-7 shows the construction of the overlay line for which the best fit is obtained (s = —17 in the minus family, as predicted), and the position of fit with the turned overlay. This overlay line is peculiar in that it abruptly reverses its trend at the point P(j-™\ with the result that Poa7) and P(2~19) fall together; this, however, is not an indication of any 178 LINKAGE COMBINATIONS [Sec. 6-4 °S fe °S § oQ I I I i o Sec. 64] EXAMPLE: DESIGN OF A THREE-BAR LINKAGE 179 peculiarity in the linkage. From the position of fit we read the following values for the constants : tW = -0.2, as > assumed, M*> = 0.025, nam = 0.200, ■X Am = -93.8°, X$M = -170°, Xzrn = -270°, B[2) If = io-°-2 = 0.630 1<2) 1<2> = 10-0-2 = 0.630, (27) Bi2) £?- = 10-0.175 = 0 66g> If Since <pi and <p2 have interchanged in this problem, Eq. (5-56) becomes <P2 = —Xz — 170° ^ = xA + 93.8°. (28) When the fitting process is carried out on a large scale it can be seen that these constants give a fit good to within 1°. To make more evident the change in constants due to this second calculation, we rewrite the above results, using Eqs. (5-68) and the obvious relations A? = gf, B[2) = 2(22), A[2) = If, BP = B{}\ (29) One finds = A? A? M&f = -0.2, ybf = 0.025, Ma(2) = 0.2, B[2) A{2) = 0.630, A? = 0.630, £<2> A[2) = 0.668. (30) To throw Eq. (28) into a form comparable to Eq. (23), one must also remember that X3 in the inverted problem corresponds to ± 180° — X4 in the direct problem, and X4 corresponds to ± 180° — X3. We have then (pi= -X3- 86.2°, (31a) <p2 = X4 + 10.0°. (316) 180 LINKAGE COMBINATIONS [Sec, 6-4 Figure 6-8 shows the mirror image of the linkage described by Eqs. (27) and (28) — that is0 the linkage described by Eqs. (30) and (31). Direct comparison can then be made with the linkage of Fig. 6-6, with respect to which this is supposed to be an improvement. The scales of Xz and X3 in these figures are mirror images, as are those of X4 and J?4, but the scales of 03 and 04, which alone are of real interest, are almost the same. It will be observed that our consideration of the inverted problem has led us back to the initially assumed value of /z&i; indeed, all constants of the linkage are essentially the same in the second approximation as they Fig. 6-8. — Mechanization of the tangent function: Second three-bar-linkage design. were in the first. This, together with the good fit obtained, might lead one to suppose that the initial choice of fxbi was unusually fortunate, that another choice would have been decidedly worse, and that the method of successive approximations would have led to convergence on the value ubi = —0.2. This, however, would be incorrect: we have here a case in which a good fit does not depend upon a particular choice of fxbi, and the method of successive approximations converges very slowly, if at all. For example, if we had chosen pb^ = —0.3 we would have found a good fit for na(l) = 0.314. Passing to the inverse problem, we would have assumed M6(,2) = -fxa'^ = -0.314 and then found M&(i2) = -jua(2) « -0.3, very closely indeed. Convergence to a definite value of fxbi is here so slow as to be undetecta- ble in a graphical method, and is not of any practical importance for obtaining a good fit. It will, however, be observed that there is one constant which is the same in all these linkages : Sec. 641 EXAMPLE: DESIGN OF A THREE-BAR LINKAGE ubi + fxa « 0.01, or log] ■do-0-01- 181 (32a) (326) It is evident that to obtain a good fit to our given function one must have the crank lengths very nearly equal. Any adjustment of parameters -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 MP Fig. 6-9. — Mechanization of the tangent function: Third approximation of the nomographic method. which tends to disturb this relation — for instance, change of a when b\ is fixed — will lead to very little improvement in the over-all fit, even though it might result in a marked improvement in some other constant of the linkage. The ideal method for adjusting constants in this problem would be one in which 6i + a could be treated as fixed and the other constants varied. This is, however, a matter of rather academic interest as the fit already obtained is quite satisfactory. This same given function can be mechanized by other radically different linkages. As noted at the end of Sec. 5-13, before accepting any design one should seek a solution of the problem with fxbi of the opposite sign — in this case positive. Trial of fib[l) = 0.2 leads to so poor a fit 182 LINKAGE COMBINATIONS [Sec. 64 that it is uncertain what value of \xa is really best. In such a case it is desirable to try another value of \xh^. We take ftb^p = 0.4. The curves of the plus family labeled s = 9 and s = 10 then give the best fit, but an intermediate overlay curve, s = 9.5, is appreciably better. Figure 6-9 Fig. 6-10. — Mechanization of the tangent linkage: Third three-bar-linkage design. shows the construction of this curve and the fit obtained on the contour fjb = 0.1. The constants of the linkage are pb? = 0.4, fib? = 0.1, jxa (i) = 0.686, Xim = 326.4° or ■X*Zm = 95°, A?> - 2.54, A™ - 4.85, = 6.10. 33.6C (33) Sec. 64] EXAMPLE: DESIGN OF A THREE-BAR LINKAGE 183 Here X$m = sd, since the fit is found in the plus family. This linkage is sketched in Fig. 6-10. By Eq. (5-56), using the upper signs, we find v>i = X3 - 95°, (34a) <p2 = X4 + 33.6°. (346) The directions of increasing 03 and 04 are then those indicated in the sketch. In applying the method of successive approximations one would normally take m&i2) = — /*a(1) = 0.686. Instead, in order to keep within the range of the nomogram, we shall take /i^i2) = 0.6. The linkage generating the inverted function must differ from that of Fig. 6-10 by reflection in a vertical line, and also in horizontal line, in order that the scale of the output quantity may increase clockwise (cf. Sec. 5-14). The fit is to be expected again in the plus family of overlay curves, for X3m m -150°, or p « — 15. It is in fact with this curve that the best fit is found, on the contour nb = 0.3. This fit is shown in Fig. 611, which makes use of the extension of the nomogram into the range 17 > 180°. The constants of the linkage as thus determined are m5(!2) = 0.6, fibp = 0.3, ^(2) = -0.435, Xim = 279.8°, X,m = 150°, R(2) Z? = 3-98' 1(2) if 4P- = 5.43. The form of this linkage, after reflection in vertical and horizontal lines, is shown in Fig. 6-12, for comparison with Fig. 6-10. By Eq. (5-56), <P2 = X3 - 150°, (36a) n - Xt - 279.8°. (366) The scales for 03 and 04, as associated with the reflected linkage, then have the senses indicated in the sketch. The linkage of Fig. 6-12 provides an excellent fit to the given function, and there is no reason to proceed to a third approximation, with /i&i3) = — m«(2) = 0.435. We have thus mechanized the given function by three-bar linkages 184 LINKAGE COMBINATIONS [Sec. 6-4 with fj.b2 < 0 and nb2 > 0, respectively. Ideally, either of these link- ages might be used. Mechanically, the second linkage is much less satis- factory than the first, both as regards space required and the magnitude of backlash error (acute angles between the cranks and the connecting links will tend to magnify backlash). In our further discussion of this example we shall therefore use the linkage of Fig. 6-8, with constants given by Eqs. (30) and (31). Direct calculation, by the methods of Sec. 64] EXAMPLE: DESIGN OF A THREE-BAR LINKAGE 185 Fig. 6-12. — Mechanization of the tangent linkage: Fourth three-bar-linkage design. Sec. 5*1, shows that this linkage generates the relation between 03 and 04 given in Table 6-3 (cf. Table 6-2). Table 6-3. — (9i\d3) as Generated by Three-bar Linkage e? degrees V(r) degrees degrees degrees d{r) gen. d[r) given 0.0 0 - 86.2 -10.10 -0.10 0.0000 0.000 0.1 10 - 96.2 - 2.80 7.20 0.0737 0.067 0.2 20 -106.2 3.08 13.08 0.1331 0.126 0.3 30 -116.2 8.71 18.71 0.1900 0.185 0.4 40 -126.2 14.66 24.66 0.2501 0.247 0.5 50 -136.2 21.48 31.48 0.3190 0.321 0.6 60 -146.2 29.88 39.88 0.4038 0.407 0.7 70 -156.2 40.80 50.80 0.5141 0.513 0.8 80 -166.2 55.17 65.17 0.6592 0.655 0.9 90 -176.2 72.56 82.56 0.8349 0.825 1.0 100 -186.2 88.90 98.90 1.0000 1.0000 With the constants as given the fit is not exact at one end of the curve, and AX 4 does not have exactly the desired value of 100°. This discrepancy could be removed by readjustment of the constants but will be corrected 186 LINKAGE COMBINATIONS 5ec. 6-5 later in an easier way. The generated 0(4r) in Column 6 of Table 6-3 is the homogeneous variable corresponding to the generated <p(2r), rather than that given by Eq. (216). The error in this quantity nowhere exceeds 1 per cent of the total travel. We have thus arrived at a first approximate mechanization of the tangent function, Eq. (13), by a combination of two ideal harmonic transformers and a three-bar linkage, with the constants Xlm = -17.5° AXi = AX3 = 100°, X2m = -70°, AX2 = AX4 = 99°, X3m= -186.2°, X4m = -10.1°, B2 (37) ?r = 0.630, 4^ = 0.630, Ai A1 = 0.668. The error in this mechanization is most easily determined by extending Table 6-3 to either side. Using the harmonic-transformer constants of Eq. (37), one can compute the values of Hi and H2 associated with the tabulated values of 03 and 04; these can be compared with values of hi and hi computed by Eq. (15). The resulting values appear in Table 6-4. Table 64. — (H2\Hi) as Generated by First Approximate Linkage hx = Hi 03 04 H2 h[0) hP - Ih 0.0000 0.0 0.0000 0.0000 0.0000 0.0000 0.1317 0.1 0.0737 0.0358 0.0328 -0.0030 0.2665 0.2 0.1331 0.0721 0.0688 -0.0033 0.4002 0.3 0.1900 0.1127 0.1103 -0.0024 0.5288 0.4 0.2501 0.1612 0.1604 -0.0008 0.6485 0.5 0.3190 0.2234 0.2247 +0.0013 0.7555 0.6 0.4038 0.3085 0.3108 + 0.0023 0.8467 0.7 0.5141 0.4300 0.4306 +0.0006 0.9191 0.8 0.6592 0.6017 0.5965 -0.0052 0.9708 0.9 0.8349 0.8136 0.8064 -0.0072 1.0000 1.0 1.0000 1.0000 1.0000 0.0000 The over-all error thus remains less than 1 per cent of the total travel. 6*5. Redesign of the Terminal Harmonic Transformers. — The methods described in Sec. 6-2 will lead one to a preliminary mechanization of the given function by a combination of a three-bar linkage and two ideal harmonic transformers. Accepting the three-bar-linkage constants as fixed, one can then improve the accuracy of the device, and at the same time bring it into a more satisfactory mechanical form, by redesign- ing the terminal harmonic transformers as nonideal. The problem of designing the two terminal harmonic transformers differs but little from that of designing a double harmonic transformer and can be solved by the same methods. (Cf. Sees. 4-13 to 4-15.) Sec. 6-6] REDESIGN OF THE TRANSFORMERS 187 Graphical Method Of Successive Approximations. — The problem is to choose operators (03|i/i) and (H2\d4), each characterized by three disposa- ble constants Xm, L, E*, which give the product operator (H2\Hi) = (#2|04) • (04|03) ' (0s\H0 (38) as nearly as possible a specified form. We first try to make (H2\Hi) identical in form with (hi\h2) by changing only one of the transformer operators — for example, (#2|04) — and assigning to (03|#i) its first approxi- mate form, (03|#i)i. The required form of (#2|04) can be determined by solution of (h2\h0 = (#2|04) • (04|03) • (03|#i): (39) by the graphical construction illustrated in Fig. 6-13 (which applies to the example discussed in Sees. 6-3 and 64). A judiciously chosen approximation to this will be (#2l04) 2. The form of (03|#i) required, in conjunction with this form of (#2|04), to make the mechanization exact, can then be determined by graphical solution of (h2\hj = (//2|04)2-(04|03)-(03|#i); (40) Next, (03|#i)2 is determined as a suitable approximation to this. (H2\6i) is readjusted, and so on until the fit can no longer be improved or until the limits of applicability of the graphical method are reached. Numerical Method. — The numeri- cal method for the design of nonideal double harmonic transformers (Sec. 4-15) can be applied to the present problem without essential change. In particular, Eqs. (4.89) to (4.97) are valid here also, provided only that H2(6z) is taken to mean the value of H2 corresponding to the specified value of 03; alternatively, we may consider i/2(03) to be an abbreviation for i/2[04(03)], where #2(04) is defined by Eq. (4-12) and the functional relation 04(03) is deter- mined by the three-bar linkage under consideration. The method will be fully illustrated in the next section. 6-6. Example : Redesign of the Terminal Harmonic Transformers. — In continuing the example of Sees. 6-3 and 6-4 we apply numerical meth- ods to the redesign of the terminal harmonic transformers. This example is of special interest in showing that straightforward application of the (#i * 2 / / / / \ / A f f / / (04 I03> i \h 2I*4 )req 1 \h2\h{) Fig. 6-13. — Graphical construction of the required form of (#2! 04). 188 LINKAGE COMBINATIONS [Sec. 6-6 method of Sec. 4-15 does not always lead to a satisfactory result; the modification required in the present case will be described. We shall keep fixed all constants of the linkage specified in Eq. (37) and shall adjust only the constants L\, L2, E*, E*, which characterize the input and output links. We shall first of all attempt to make the error in the mechanization vanish for 03 = 0.2, 0.4, 0.6, 0.8. In Table 6-5 will be found the quanti- ties needed to give explicit numerical form to Eqs. (4-97). Values of 03 and 04 will be found in Table 6-4. Values of Hf can be found from Table A*l by an interpolation between corresponding entries in the columns AXi = 100°, Xim = -20°, and A_Y< = 100°, Xim = -15°. To obtain values of H* would require interpolation in both AX» and 04 ; it is advisable to make a direct calculation by Eq. (4-42). The values of Hf and H$ in Table 6-5 have been thus obtained. The f's have been computed from the H's and H*'s, and the values of dh2/dhi have been computed as /dhj\ \dhJhl=Hl 9 tan 80c sec2 (80° • Hi). (41) Table 6-5. — Constants Required in Design Procedure 03 Ht /i<».) /■<*) H* Mo,) fM dfo din 0.0 0.9468 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.2 0.9989 0.3403 -0.6087 0.3120 0.0501 -0.5073 0.2837 0.4 0.9124 0.4101 -0.9325 0.5505 0.1974 -0.8400 0.4501 0.6 0.6978 0.2678 -0.9325 0.7962 0.4318 -1.0930 1.0115 0.8 0.3809 0.0726 -0.6087 0.9848 0.5757 -0.9956 3.0622 0.9 0.1957 0.0121 -0.3361 0.9630 0.3944 -0.6090 5.399 1.0 0.0000 0.0000 0.0000 0.8094 0.0000 0.0000 Equations (4-97) become, for 03 = 0.2, 0.4, 0.6, 0.8, 0.9, respectively, -0.0965a + 0.1727b + 0.0501c - 0.5073d = -0.0033, (42a) -0.1846a + 0.4197b + 0.1974c - 0.8400d = -0.0008, (426) -0.2709a + 0.9432b + 0.4318c - 1.0930d = 0.0023, (42c) -0.2223a + 1.8640b + 0.5757c - 0.9956d = -0.0052, (42d) -0.0653a + 1.8146b + 0.3944c - 0.6090d = -0.0072. (42e) Attempting to reduce the error to zero at the first four values of 03, we solve simultaneously the first four of these equations, and obtain a = 0.0246, b = -0.0206, c = 0.0702, d = 0.0017. (43) Equation (4-31) gives gx = 0.6751, g2 = 0.4627. (44) Sec. 6-6] REDESIGN OF THE TRANSFORMERS 189 By Eqs. (4-46) and (4-47) we have then 0.1070 0.2279 In = — - — , L2 = F* -- E* = (45) or U = 9.264, L2 = 1.525, Ef = -0.837, E* = 0.024. (46) Calculation, by the methods of Table 4:5, of values of H{, H2, and of h20)f (the value of h2 corresponding to hi = H[), yields the results shown in Table 6-6. The over-all error, h'20) — H2, of this mechanization is actually larger than that with which we started, rather than zero at the chosen values of 03. This is evidently due to excessively large errors in the approximate linear equations used in the design procedure. In order Table 6-6. — Pebformance of the Linkage [Eq. (45)] Bz Hi Hi H2 K MM K" M0)' - H'2 0.2 0.2665 0.2889 0.0721 0.0746 0.0688 0.0753 0.0007 0.4 0.5288 0.5602 0.1612 0.1732 0.1604 0.1752 0.0020 0.6 0.7555 0.7831 0.3085 0.3362 0.3108 0.3409 0.0047 0.8 0.9191 0.9344 0.6017 0.6398 0.5965 0.6469 0.0071 0.9 0.9708 0.9785 0.8136 0.8630 0.8064 0.8499 0.0100 to make a small correction in the over-all generated function we have been forced, by its peculiar form, to use harmonic transformers that deviate strongly from the ideal form ; h20) — U2V and H 2 — H2 are large. Accord- ing to our approximate equations, these large corrections should nearly cancel, leaving an over-all correction of much smaller magnitude and of the desired form. Unfortunately, in computing these large corrections with the linear equations we have made errors that do not tend to cancel out — errors that, in their aggregate, are even larger than the difference that it was desired to compute. Accordingly, the expected accuracy in the correction has not been realized. Difficulties of this type can sometimes be avoided by very slight modifications in the conditions imposed. In the present case, for instance, one need admit only a very small error at 03 = 0.4 in order to use harmonic transformers that are more accurately described by the linear equations; the linkage thus designed has a performance much closer to one's expecta- tions, and correspondingly more satisfactory. We desire to make a positive correction at 03 = 0.2, a negative one at 03 = 0.6. It is evident, then, that the correction made will tend to be small at 03 = 0.4, whether or not special care is taken with this point. We shall therefore release this point from direct control, and shall solve the first, third, and fourth of Eqs. (42) for the constants a, c, d, in terms of 190 LINKAGE COMBINATIONS [Sec. 6-6 the constant 6. (The choice of the constant 6 for this special treatment is quite arbitrary.) One^finds a = -0.226 847 - 11.400 571 b, c = -0.015 059 - 3.980 738 b, d = 0.048 170 + 2.115 946 6. (47a) (476) l47c) The error to be expected at 03 = 0.4 is 0.0008 plus the quantity on the left-hand side of Eq. (426) : do. -0.00076 - 0.03895 6. (48) As one would expect, 50.4 is quite insensitive to the choice of 6; we can choose this constant with the idea of getting a good mechanical design, well described by the linear equations. We desire, then, that L\ and L2 shall not be either very large or very small, and that E* and E* shall he between zero and one. It follows that a and c should be of the order of magnitude of 0.1, that 6 should have the same sign as a, and that d should have the same sign as c. We can give L\ and L2 roughly equal magni- tudes, and obtain the desired sign relations, by setting 6 = —0.0135: a = -0.07294, 6 = -0.0135, c = 0.03868, d = 0.01960, (49) Li = -3.1245, L2 = 2.7676, Ef = 0.185, E% = 0.507. The expected value of 50.4 is then —0.0002. The actual performance of the linkage is shown in Table 6-7. Table 6-7. — Performance of the Linkage [Eq. (49)] 03 H[ H't h[0)' - H'2 0.0 0.0000 0.0000 0.0000 0.1 0.1217 0.0302 0.0001 0.2 0.2504 0.0640 0.0003 0.3 0.3819 0.1036 0.0005 0.4 0.5120 0.1523 0.0008 0.5 0.6361 0.2159 0.0010 0.6 0.7489 0 . 3037 0.0006 0.7 0.8457 0.4293 -0.0003 0.8 0.9219 0 . 6045 0 0005 0.9 0.9743 0.8170 0.0086 1.0 1.0000 1.0000 0.0000 The errors due to use of the approximate equations are small, and the performance of the linkage is satisfactory at the points controlled. Unfortunately, the error increases rapidly for 03 > 0.8, and the design can not be considered acceptable. It is evident that in the design process more attention must be paid to the error for 63 = 0.9. Sec. 6-6] REDESIGN OF THE TRANSFORMERS 191 An attempt to control the error at 03 = 0.9 instead of 03 = 0.8, by using Eq. (42e) instead of Eq. (42c?), leads to similar results: one can actually make the error at 03 = 0.9 very small, but the error at 03 = 0.8 takes on a large negative value. An attempt to make the error vanish for both 03 = 0.8 and 03 = 0.9, by solving simultaneously Eqs. (42a), (42c), (42d), and (42e), leads to calculation of the constants Li = 1.030, 0.771. Ef = -0.1-71, E% = -0.250. (50) These values are such that the linear equations can not be expected to be accurate; the linkage will not give the expected good performance even at 03 = 0.8 and 03 = 0.9. The case here encountered is in fact one in which adjustment of the constants L\, L2, E*, E* can not bring the over-all error within very strict tolerances, such as ±0.001. Readjustment of Xim and X2m, or even a new resolution of the given function and redesign of the three-bar- linkage component, would be required if such accuracy were demanded. On the other hand, a tolerance of ±0.0025 can be met without such redesign, by a somewhat different approach. Our problem is to correct the error appearing in the last column of Table 6-4 by making the proper linear combination of four correction functions: - 0^\ /i(03), - (~A /2(03), /,(*,), and/4(03). We have been dealing with special values of these functions as coefficients in Eqs. (42). These are reproduced, with the error to be corrected, in Table 6-8. What is required is that we make linear combinations of Table 6-8. — Error Correction Functions dz dh-z , dh 'Jl dh2 , ~dh'h /a U /i<0) - H2 Fx F, 0.2 0.4 0.6 0.8 0.9 -0.0965 -0.1846 -0.2709 -0.2223 -0.0653 0.1727 0.4197 0.9432 1.8640 1.8146 0.0501 0.1974 0.4318 0.5757 0.3944 -0.5073 -0.8400 -1.0930 -0.9956 -0.6090 -0.0033 -0.0008 0.0023 -0.0052 -0.0072 -0.0469 -0.0641 0.0000 0.3130 0.4559 -0.1503 -0.1344 0.0000 0.1824 0.1538 entries in Columns 2 to 5, inclusive, with coefficients a, 6, c, d, such that the sums approximate as well as possible to the corresponding entries in Column 6. Examination of Table 6-8 will make it clear that our diffi- culties have arisen from the attempt to make a positive correction in the center of the range and a negative correction at both ends, whereas not one of the error-correction functions changes sign. The error at 03 = 0.6 can be tolerated; let us therefore make no correction at this point and attempt only to reduce the errors at the ends of the range. We shall in fact design 192 LINKAGE COMBINATIONS 5ec. 6-6 the input and output transformers so that neither changes the generated function at 03 = 0.6, taking = E* = 0.2872, = E* = 0.3950. (51) The form of the correction made in each terminal transformer (Fi and F2 in Table 6-8) is thus fixed; it remains to determine the constants a and c with which these should be added. Next, let the (approximate) error at 03 = 0.2 be required to vanish: -0.0469a - 0.1503c = -0.0033. The errors at 03 = 0.8 and 03 = 0.9 will then be (52) 50 8 = -0.0052 - 0.3130a - 0.1824c = -0.0092 - 0.2561a, (53a) 50.9 = -0.0072 - 0.4559a - 0.1538c = -0.0106 - 0.4079a. (53b) It is evident that for best results one must use a negative a, and allow a negative error at 03 = 0.8, a positive one at 03 = 0.9. An appropriate choice is a = -0.0306, c = 0.0315. (54) We thus find as constants for the linkage Lx = -7.438, U = 3.398, E* = 0.2872, E* = 0.3950. (55) The performance of this linkage is shown in Table 6-9; it is about the best that can be attained by adjustment of these four constants. Table 6-9. — Performance of the Linkage [Eq. (55)] 03 H[ K K0)' - H\ 0.0 0.0000 0.0000 0.0000 0.1 0.1285 0.0324 -0.0004 0.2 0.2615 0.0674 0.0000 0.3 0.3948 0.1077 0.0007 0.4 0.5244 0.1570 0.0015 0.5 0.6461 0.2209 0.0023 0.6 0.7555 0.3085 0.0023 0.7 0.8487 0.4334 0.0008 0.8 0.9222 0.6075 -0.0015 0.9 0.9734 0.8185 0.0021 1.0 1.0000 1.0000 0.0000 Sec. 6-7] ASSEMBLY OF THE LINKAGE 193 6*7. Example: Assembly of the Linkage Combination. — The final step in the mathematical design of a linkage combination is to coordinate properly its component parts. Careful attention must be paid to sign conventions and to the varying zero lines from which angles are measured in the several types of components. It is safest to begin by sketching the component linkages in their basic positions. With each component there should be indicated the scales for the output parameters. In our example the linkages and scales are fully characterized as follows: Input harmonic transformer: Xta = -17.5°, Li -7.438, AXi = 100°, EX = 0.287, 02 increases with Xi. !-bar linkage : XZm = -186.2°, 41 - 0.630, AX3 = 100°, 4* = 0.630, Ai xim= -10.1°, ^ = 0.668, Ai AX4 = 99°, 100° • 03 = 86.2° - X3, 99° • 04 = 10.1° + X4. Output harmonic transformer: X2m = -70°, U = 3.398, AX2 = 99°, E$ = 0.395, 04 increases with X2. These linkages are sketched in Fig. 6-14; scales of Hi, H2, 03, and 04 are shown. There is an adjustable scale constant in the design of each com- ponent. The scale constants of the harmonic transformers (a and c, respectively) can be adjusted to control the travels at the input and out- put terminals; choice of the scale constant of the three-bar linkage (6) is subject only to considerations of mechanical convenience. In the linkage combination the readings on the 03-scales of the input harmonic transformer and the three-bar linkage must always be the same. We have designed the two 02-scales to cover the same angular range, but the sign conventions have forced us to allow the 03-scale of the three-bar linkage to increase counterclockwise, whereas that of the transformer increases clockwise. These components might, for instance, be con- nected by the gearing indicated in Fig. 6-14. On the other hand, the 04-scales of the three-bar linkage and the output transformer increase 194 LINKAGE COMBINATIONS 17.5! [Sec. 6-7 H2=0.0 0 I i Fig. 6-14. — Components of the tangent linkage. Fig. 6-15. — Completed linkage mechanizing X2 = tan xi, 0 < Xi < 80°. Sec. 6-8] THE DOUBLE THREE-BAR LINKAGE 195 in the same sense; one possible method of connecting these components is indicated in Fig. 6-14. This completes the preliminary representation of the linkage combination. Finally, one must convert the preliminary representation into a prac- tical design without changing the essential relations of the components. One possible arrangement of this tangent linkage is shown in Fig. 6-15. The three-bar-linkage component is rotated through 90° from its position in Fig. 6-14, largely to gain clarity in the representation. The output crank of the three-bar linkage and the crank of the output transformer are made to rotate together as arms of the same bell crank. In order to use the same type of connection between the input transformer and the three-bar linkage, we must reverse the sense of rotation of one or the other of these cranks. This can be done by reflecting the input transformer and its associated scales in a vertical line. The two cranks can then be joined into a bell crank, and the linkage appears as in Fig. 6-15. THREE-BAR LINKAGES IN SERIES It is not desirable to use harmonic transformers in a computer in which all variables are represented by shaft rotations since the linear motion of the input or output slides must then be transformed into a rotary motion by a rack and pinion ; it is much better to take the rotary motion directly from a rotating terminal. This remains true even when the angular travel must later be increased since this can be accomplished by gears that permit a more compact design than the rack and pinion. For such computers a single three-bar linkage is ideal, except that it does not permit generation of a sufficiently large class of functions to cover all practical cases. Systems of two or more three-bar linkages provide greater flexibility, together with the same satisfactory mechanical characteristics. 6-8. The Double Three -bar Linkage. — In a double three-bar linkage, such as that sketched in Fig. 6-16, the homogeneous input parameter 0i is transformed into an intermediate parameter 03 by the first three-bar linkage; this serves as the input to the second three-bar linkage, which generates the output parameter 02. In the operator symbolism, (02|0i) = (02|03) • (08|0i). (56) The three-bar-linkage operators are each characterized by five constants, (AXi, AX3, &n, 621, fli) and (AX3, AX2, &12, b22, a2), respectively. Since the linkages must have a common value of the constant AX3, the number of disposable constants in the combination is nine. The design problem is to choose operators (02|03) and (03|0i) such that their product (02|0i) approximates as closely as possible to the given function h2 = Qi2\hi) • hi (57) 196 LINKAGE COMBINATIONS [Sec. 6-8 on direct or complementary identification of the variables 6h 02 with the variables hi, h2. Formally this problem resembles closely that of designing a double harmonic transformer, and the general approach to it is the same. For in- stance, one can apply the method of successive approximations described in Sec. 4-13. In each stage of the pro- cedure one must then fit a three-bar- linkage function of specified AX3 to a known function by an application of the nomographic or geometric method. Aside from this increase in manipu- lative difficulties, the principal difference between this problem and the earlier one. lies in the first step, in which one must make an initial Fig. 6-16. — Double three-bar linkage. \ 37°42'58; 36*05' Fig. 6-17. — Constants of a double three-bar linkage mechanizing the logarithmic relation for 1 < x < 50. choice of one of the factor operators. It is to be noted that this choice fixes a value of AX3 which will be used throughout the design procedure. Sec. 6-8] THE DOUBLE THREE-BAR LINKAGE 197 One can begin by using an operator (0i|0s), for example, which by itself gives a rough fit to the given operator; the second factor will then serve to make relatively small corrections. This procedure leads to the design of combinations of quite different linkages, such as that illustrated in Fig. 6- 16. A generally sounder procedure is to try to find a combination of more or less similar linkages which will make roughly equal contributions to the curvature of the generated function (cf . Fig, 6-17) . An appropriate begin- Fig. 6-18. — A possible physical form for the logarithmic linkage. ning is then made by factoring the given operator into the product of two identical operators W: (h2\h!) = W'W = W'< (58) The operator W, called "the square-root operator," has been discussed in Chap. 3, where it has been shown that it is not uniquely determined. If any one of the square-root operators can be mechanized by a three-bar linkage with equal input and output travels, then two of these linkages in series will generate the given function. If only an approximate mecha- nization of W can be found, the corresponding operator can at least serve as a first approximation to (0s|0i), with which to begin application of the method of successive approximations. 198 LINKAGE COMBINATIONS [Sec. 6-8 An example of a linkage obtained by use of the square-root operator W is provided by a patented linkage1 mechanizing the relation x2 = log™ xh 1 ^ Xi ^ 50, (59) with an error everywhere less than 0.003 of the output travel. In design-" ing this, a three-bar linkage was used to mechanize, with good approxima- tion, that one of the square-root operators W which has derivatives at the ends of the domain. Two such linkages in series gave a mechanization of the given function which was sufficiently good to permit immediate application of the methods of Chap. 7 in a final adjustment of the con- stants of the linkage combination. The final linkage is shown sche- matically in Fig. 6-17; the component linkages have similar, but not identical, constants. The angle (3 of the combination can be chosen at will. Figure 2-10 shows the linkage obtained on setting /? = —142° 34'. A mechanically preferable form is that shown in Fig. 6-18, in which the two linkages share the intermediate crank : /3 = 0 and 2.37259a = 1.801246. 1 A. Svoboda, U.S. Patent 2340350, Feb. 1, 1944. CHAPTER 7 FINAL ADJUSTMENT OF LINKAGE CONSTANTS 7*1. Roles of Graphical and Numerical Methods in Linkage Design. — The preceding chapters have been concerned with methods for linkage design that are largely graphical, rather than numerical. Graphical methods are easily applied, and have the important virtue of making evident the character of the over-all fit to the given function, not merely the fit at a selected set of points. Their accuracy, however, is limited; when high accuracy is required, the final adjustment of linkage constants must be carried out by numerical methods because these alone permit sufficiently careful adjustment of the constants and sufficiently accurate evaluation of the performance of the linkage. Numerical methods, on the other hand, tend to be excessively complex, except when they relate to changes in linkage constants so small that one can assume that the error function depends linearly on each of these changes. Graphical methods are thus very important in making it possi- ble to find, quickly and easily, a linkage with constants which need to be changed only a little to bring its structural errors within the specified tolerances of the problem; it is only at this point that numerical methods become effective and convenient. In general, then, graphical methods are desirable for the first stages of linkage design, which must yield a linkage with small error over the whole range of travel. The error can then be further reduced by numeri- cal methods; often it can be made to vanish at several selected points. This was, for instance, the method employed in Sees. 4-7 and 4-15. The present chapter will provide a general discussion, for linkages with one degree of freedom, of the problem of making final adjustments of all disposable constants of a linkage. It will be a basic assumption that the structural error at any point is nearly a linear function of each of the variations of constants to be considered. Thus the discussion will in most, but not all, cases apply only to small changes of the constants. Sometimes these methods are convenient even when an improved basic outline of the system is to be obtained by a substantial change in some constant. Such may be the case when the graphical method has been so applied that it does not establish a near optimum design within a whole class of linkages — for instance, when a combination of a three-bar linkage and harmonic transformers has been designed with frozen angular travels, 199 200 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-2 and one must consider the possibility of making fairly large changes in these travels. The chapter will conclude with a discussion of a quite different method of reducing structural errors, which is particularly useful after the usual numerical methods have been applied: the introduction of small cor- rections by the eccentric linkage. 7*2. Gauging Parameters. — Let us consider the problem of checking the performance of a linkage designed to mechanize a given relation between variables Xi and x2: x2 = (x2\xi) ■ Xi. (1) The linkage will generate a relation between an input parameter Xi and an output parameter X2: X2 = (X,|X,) • Xi. (2) The form of the operator (X2\Xi) will depend upon dimensional constants of the linkage, the precise nature of which we need not specify. We denote these by g0, gh g2, . . . gn-4. At the input terminal there will be a linear scale which relates the values of the input variable and the input parameter: Xl = Xj°> + hfci - s£°>), (3) X£0) and x^ being corresponding values of these quantities. At the out- put terminal there will be a similar scale relating the output parameter to the actually generated (not the ideal) values of the output variable. Denoting by x2a these actual output values of the mechanism, we have X2 = Xp + k2(x2a - sB), (4) X20) and XJJ again being corresponding values. The linkage and scales together generate a relation between x\ and x2a, which depends on the constants of the linkage and on the four additional constants, ki, X^\ k2, and X(20), which characterize the terminal scales. These latter constants we denote also by gn~z, Qn-2, gn-i, and gn. We have then x2a = F(xh gQ, gh • • • gn), (5) a function of the input variable and n + 1 constants of the mechanism. Perhaps the most obvious way to study the structural error of the mechanism is to compare the desired and the actually generated values of the output variable for a spectrum of values of the input variable, ^(0) ~(1) ^(r) The corresponding spectrum of values of x2 is determined by Eq. (1) : x2s) = (xt\xi) ■ x{8); s = 0, 1, • • • r. (6) Sec. 7-3] USE OF THE GAUGING PARAMETERS 201 Similarly, Eqs. (2), (3), and (4) determine spectra of values of Xh X2} and x2a. In particular, x<& = F(x[8\ g0} pi, • • • <7«); s = 0, 1, • • ■ r. (7) The structural error, dx2, of the mechanism has the spectrum &4S) = xfa - s</>; s = 0, 1, • • • r. (8) The corrections which one would like to make in the output of the mecha- nism are the negative of these quantities. In such a test a comparison of the ideal and the actually generated values of x2 is used as a gauge of the precision of the linkage; we shall say that x2 is used as the gauging parameter. It is not at all necessary to use x2 in the gauging process. In most cases this is not even desirable ; it is better to use as a gauging parameter one of the dimensional constants of the linkage, go, <7i . . . <7«. Let us solve Eq. (7) for this gauging parameter, for instance go: go = G(x[a), x(&, glf gh • • ■ gn). (9) If we substitute on the right any corresponding values of X\ and x2a, we shall compute always the same value of go — the actual value of this con- stant in the linkage considered. If, however, we use ideal values of x2, x28), instead of the actually generated values, x^l, go will not in general have a constant value, but instead a spectrum of values, <#> = G{x\*\ *%>, glt g2, - • • gr„); s = 0, 1, 2, • • • r. (10) The difference between the actual value go of the constant and the value g(08) which it would need to have to make the linkage exact at the point s we shall call the gauging error, ¥os) -V- 9tf, s = 0, 1, ■ • • r. (11) Such quantities are useful as gauges of the precision of linkages, although they do not give directly the error at the output. A wisely chosen gauging parameter is usually simpler to calculate and easier to interpret (at least as regards desirable changes in the constants) than is the error at the output; in particular, if 5g(0s) is independent of s it is only necessary to reduce g0 by this amount to make the linkage exact. That the proof of perfect performance of the linkage is reduced to demonstration of the constancy of the results of a series of computations is also of value for the avoidance of computational errors. 7»3. Use of the Gauging Parameter in Adjusting Linkage Constants. — In the preceding chapters we have seen how to design linkages with small gauging errors. It may still be desirable to improve these linkages by making small variations in the dimensional constants <7o, <7i, £2, . . • gn. 202 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-3 A perfect linkage will be obtained if values of gh . . . gn, can be found such that 0(os) as computed by Eq. (10) is the same for all possible sets of values (x\8), x2s)). In general, one can at best hope to make g0 constant at a preassigned set of points equal in number to the independent con- stants of the linkage and thus to obtain a linkage which generates the given function exactly at these points. If the dimensional constants are changed by amounts Agi, becoming g'i = gi + Agiy i = 0, 1, • ■ • n, (12) the gauging parameter will have the spectrum of values &» = G&p, xf, g[, gf2, • • ■ g'n), (13) and the gauging error will become ^ = g'0- g'o(s\ s = 0, l, • • • p. (14) Expanding Eq. (13) in a Taylor's series, we may write g'^ = ffp + Agtf = gV + ^ 1f%< + 2! Li Lf * -■' A9iA9i + ' ' ' ' the partial derivatives being evaluated at (x(i\ x(2s), gi, g2, . . . gn). The gauging error can thus be written as Sg'o(s) =9o + Ago - g(Qs) Zdg(0s) A 1 V V dW a a "ft*" -2 2, 2, «&**»- (16) It is desired to reduce this to zero at a chosen set of p + 1 precision points: s = 0, 1, 2, • • • p. The general solution of this problem is prohibitively difficult, and it is necessary to make an approximation which will be valid only if the required changes in the constants are sufficiently small. Terms in Eq. (16) of higher than the first order in the small quantities Agi will be neglected. Let Gw =M1, cj.) 1. (17) ogi Then, by use of Eq. (11) one can rewrite Eqs. (16) thus: Gl»Agi = 5g?; s = 0, 1, • • • p. (18) I Sec. 7-3] USE OF THE GAUGING PARAMETERS 203 A set of Agjs which solve these equations will serve as corrections to the originally chosen gi, as indicated in Eq. (12), under restrictions which must be discussed. One can solve this system of linear equations for the Ag{ if the ranks of the matrix of coefficients [G^s)] and the augmented matrix ([G^s)] with the added column 5g(0s)) are equal. In less precise but more direct terms, the equations will usually be soluble if and" only if the number of inde- pendent constants characterizing the generated function is equal to or greater than the number of equations, p + 1. It would be natural to infer from this statement that the linkage can be made to generate a given function exactly at m arbitrarily chosen points whenever the generated function is characterized by m mathematically independent constants, (m ^ n + 1). In practice it will be found that this is not the case; the number of precision points which can be obtained depends upon the nature of the linkage and the given function, and on the way in which the precision points are chosen. Even when the linkage under considera- tion is well adapted to generation of the given function one must often be content to fix fewer than m precision points, or to use other methods of reducing the error. This difference between the mathematical problem of solving Eqs. (18) and the practical problem of finding a linkage with p + 1 precision points arises from the fact that Eqs. (18) are mathematical approximations valid only for sufficiently small Agi. For practical purposes one must not only solve Eqs. (18), but must solve them with Agi which are so small that the quadratic terms in Eqs. (16) are negligible. We have seen in Sec. 6-6 (see Table 6-6) how different may be the expected and the actual per- formance of a linkage designed by using approximate linear equations very similar to Eqs. (18), when the Agi are so large that neglected terms are important. Difficulties are most likely to arise in the straightforward application of Eqs. (18) when the restriction to small Agi has, for practical purposes, the effect of establishing a relation between mathematically independent parameters. To simplify the discussion of this point we shall assume that the parameters gi(i = 0, 1, • • • n) which occur in this equation are all inde- pendent of each other. One can then attempt to fix n + 1 precision points, determining the Agi by solving n + 1 of Eqs. (18). Because of the independence of the parameters, the determinant of the coefficients G\s) will not vanish; the solution for the Agi will be uniquely expressible as a fraction in which the numerator is the determinant \G^S)\ with one column replaced by the column of coefficients 8g(0s\ and the denominator is the determinant \G$s)\ itself. The smaller the gauging errors 5g(0s) the smaller will be the Ag^ However, even when the 8g(Qs) are very small it may be 204 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-3 found that the Ag* are large and that the linkage with constants given by Eq. (18) does not have the desired precision points, or even an improved performance. This happens most frequently when the determinant IG^I, although not exactly zero (as it would be if there were an exact relation between the parameters), is very small. In such cases one can make large and properly related changes in the parameters which produce only a small net change in the generated function. For instance, as illustrated below, it may be possible to make large changes in two param- eters, gi and a,-, which will change the generated function very little if gi/gj is held constant. When one restricts attention to small changes in the parameters the generated function then depends, in effect, on a smaller number of parameters; in our example it would depend, not on gt and g}- individually, but only on gi/gj. Thus the number of effectively independent parameters may be decreased by restricting considerations to small Agi} and with it the number of precision points which one can hope to establish. An almost trivial example is illustrated by Fig. 7-1. A single pivoted arm (Fig. 7- la) can be used in the generation of linear functions. The mechanism itself involves no adjustable constant. The input scale is characterized by the two parameters kx and X(i\ the output scale by the two parameters k2 and X2°\ The generated linear function is characterized by only two independent constants; equal changes in X{0) and X(20) or proportional changes of all four variables do not produce any change in the generated function. In using such a device as a mechanization of an almost linear function one cannot in general reduce the error to zero at more than two preassigned points. Now consider the three-bar linkage in Fig. 7-16. It is almost a parallelogram linkage, and generates an almost linear relation between x\ and x2 — one which is characterized by seven mathematically independent parameters. The determinant \Gl8)\, with seven rows and columns, will not vanish; it will, however, be very small, and vanish as the parallelogram condition, Bi — A2, B2 = Ah is attained. It is obvious that equal changes of X[0) and X(20) will produce very small changes in the generated function, and that proportional changes of kh k2, X(!0) and X20) will have a similarly small effect. Conversely, certain small changes in the generated function will be obtainable only by making such (b) Fig. 7-1. — (a) Mechanization of a linear function. (6) Mechan- ization of an almost linear function. Sec. 7-5] LARGE VARIATIONS OF DIMENSIONAL CONSTANTS 205 large changes in these parameters that the linear theory will not apply. If we exclude large Agi from consideration there are in effect two fewer degrees of freedom than one might expect ; an attempt to establish seven precision points will be likely to fail, although five should be obtainable if the initial fit is good. Such is usually the case with three-bar linkages, which will receive more detailed discussion in Sec. 7-7. 74. Small Variations of Dimensional Constants. — It is usually desir- able to apply the approximate linear form of the gauging-parameter method, even when one must reduce the number of precision points in order to deal with small variations of the dimensional constants. Let the number of independent dimensional constants be n + 1, and the number of specified precision points be k less than this. It is then possible to solve any n + 1 — k independent equations from among Eqs. (18) for any n + 1 — k of the A^'s, in terms of the other k of these quantities. For instance, solving for AgQ, Agh . . . Agn_k, in terms of A<7n_fc+i, . . . Agn, one obtains relations of the form Ag0 = Coo + CoiAgn-k+1 + • • • + CokAgn, \ Agi = Cio + CnAgrn_fc+i + • • • + CikAgn, I . Agn-.k = Cn-k,o H~ Cn—k,iAgn-k+ 1 + * * * + Cn-k,kAgn. J Any set of small Ag's which satisfies Eqs. (19) will constitute a valid and practically useful solution of the given problem. Such a solution is not mathematically unique, but it will be effectively so if one is attempting to establish the maximum number of precision points subject to arbitrary choice. 7*5. Large Variations of Dimensional Constants. — By solution of Eqs. (18) one can determine a set of changes Ago, Agh . . . Agn, in the dimensional constants which reduces to zero the first-order terms in the gauging error. Equations (16) become, exactly, n n t'=l j=l When this gauging error is negligible we say that the A^'s are small; the problem has been solved in the first step. We turn now to the case in which 8g'0(s) is not negligible, but is adequately represented by the second- order terms written out in Eq. (20) . In such cases the design problem is not satisfactorily solved by a first application of the method of Sec. 7-3, but it can often be solved by successive applications of the method, which produce successive improvements in the dimensional constants. We shall now see how the convergence of this process can be hastened. 206 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-6 Knowing the Ag's which solve Eqs. (18), and the gauging error 8g'J8) after these corrections are made, one can easily compute also the gauging errors resulting when proportionally larger or smaller changes are made in the dimensional constants. Let a new set of corrected dimensional constants be given by g"W = 9i + AAfl, i = 0, 1, • • ■ p. (21) In the Taylor's series expansion for the gauging parameters, Eq. (15), the first-order terms are then changed by a factor X, the second-order terms by a factor X2, and so on. The resulting gauging error is n n n i=l 1=1 j =1 + • ' • , (22) or. by application of Eqs. (12), (18), and (20), Vo(w = (1 ~ Wos) + XVo">, s - 0, 1, • • • p. (23) The validity of this expression of course depends on the possibility of neglecting higher-order terms in Eq. (20). When X = 0 the dimensional constants and gauging error have their uncorrected values. Increase in X will reduce the gauging error 8g"(8) so long as the quadratic term in Eq. (23) remains negligible. As X approaches 1 the quadratic term will eventually (by our assumptions) become appreciable, and may even become very large. It is evident that one can obtain a smaller gauging error by applying a fraction of the correction indicated by the linear theory (0 < X < 1) than by applying the whole correction (X = 1). The more important the quadratic terms the smaller will this fraction be; it is, however, always possible to find some positive value of X which gives a better set of constants than either X = 0 orX = 1. In practice one begins with knowledge of 8g(0s\ and computes the Ag's. As a check on the validity of the calculation one should then determine the values of dg'o(8), usually by direct calculation [Eqs. (13) and (14)] rather than by use of Eqs. (20). If these quantities are not satisfactory small, one should make a smaller change in the g's; the appropriate value of X can be determined by use of Eq. (23), X being chosen to make the quantities 8g"(s) as small as possible. The constants g" computed by Eq. (21) will then serve as initial values for a second application of the method. 7-6. Method of Least Squares. — The designer's ultimate objective is to assure that the output error 8x2 = x2a — x2 (24) Sec. 7-7] GAUGING THE THREE-BAR LINKAGE 207 shall be kept small throughout the domain of operation of the mechanism. One way to assure this is so to choose the dimensional constants of the mechanism, on which 8x2 depends, as to minimize the integrated squared error, I(9o, 9h 9z, ' ' ' 9») - J (8x2)2dx2> (25) or the corresponding sum over a discrete spectrum of output values, E (go, ?i,--- g«) = ^ W)2-. (26) r Such conditions are most reasonable when accuracy is equally important for all values of the output variable, or all values of r. More generally, one should introduce a weighting function, w(x2) or w(r), which increases with the importance of accuracy in the result at the corresponding x2 or r. One will then so choose the g's as to minimize -/ w(x2) 8x2]2 dx2, (27) or Ew = Yl Mr) hx<??> (28) r subject to any other conditions which must be imposed on the dimen- sional constants. Least-squares methods suitable for use in linkage problems have been developed by K. Levenburg.1 It is, however, the opinion of the author that least-squares methods are relatively unrewarding. In particular, when the method depends on the use of an expansion in which only linear terms are retained there is always the danger that a result obtained after a large expenditure of labor may be invalidated by this approxi- mation. In general the author prefers to set tolerances on the output error — tolerances which may vary with x2 or r — and to apply the methods of the preceding sections to bring the actual structural errors within these tolerances. 7«7. Application of the Gauging-parameter Method to the Three-bar Linkage. Formulation of the Equations. — In applying the gauging-param- eter method to three-bar linkage design we may choose the dimensional constants as follows: 1 K. Levenburg, "A Method for the Solution of Certain Nonlinear Problems in Least Squares," Quart. Appl. Math., 2, 164 (1944). 208 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-7 go -a)' 0i = B, (29) gz = X[», g* = A-i, 05 = X20), As gauging parameter we shall use go. In effect, we shall gauge the error of a design by computing the required length B28) of the connecting bar as a function of the other dimen- sional constants and the variable pairs (x¥\ x28))', we shall seek to make con- stant the related gauging parameter, *-(#■ (30) Fig. 7-2. — Three-bar linkage. rule, symbolized by x[8) ■■ = W}|s) • s. The spectrum of x2 is then deter mined : r(«> - •«/2 — : fel^i) ' x[8> Equations (3) and (4) become X? = gz + g*{x[» + <7eW> A spectrum of values of Xi can be chosen according to some arbitrary (31) (32) 40)). (33) (34) Let the horizontal separation of the ends of the connecting bar, in terms of the unit Ah be Uia), and the vertical separation, in the same units, be V{8). Then by the geometry of the linkage (Fig. 7-2) we have [/(*) = 1 + gj Cos X[8) - g2 cos X28\ F« = gi sin X{8) - g2 cos X28\ <#) = (jjwy + (7W)!. (35) (36) (37) An equation of the form of Eq. (9) could be obtained by eliminating from Eqs. (33) to (37) the quantities X[8), X?, U^8\ and V(8\ This, however, is not necessary for our purposes. Sec. 7-8] GAUGING A THREE-BAR LINKAGE 209 The partial derivatives G[» = ^ (38) will now be given in a form suitable for numerical calculation : G?> - -1 =Qi>8) (39) i gis) = y(s) sin Xf + C/w cos Xf = Q[8) (40) - %Qf = 7<-> sin Xf + t/<s) cos X<s> = Q(2S> (41) 1 ^ = yw cos Xf> _ ij(s) sin Xp = Qy> (42) 2<7i 1 £<«) = J- £?>(*?> - zf ) = <2(4S) (43) 47i *0i _1 2<?2 _ _Lg(5s) = F<8) cos X(28> - £/(8) sin X(28) = Q(b8) (44) -i^^-i^)^-^ -«•• (45) It is to be remembered that all angles are expressed in radians, and that gr3, 04, 06, and 06 must be interpreted correspondingly. One can use the quantities Qis) directly in the solution of Eqs. (18). On introduction of the quantities Aqi = # ' A9i' * - 0, 1, • • •; n, (46) which are simply constant multiples of the A0's, Eqs. (18) become n V QWAqt = dfl>, s = 0, 1, • • • p. (47) Having solved Eqs. (47) for the Ag's, one can compute the A0's by Eqs. (46). 7*8. Application of the Gauging -parameter Method to the Three-bar Linkage. An Example. — As an example of the gauging-parameter method we shall check and improve the logarithmic linkage designed by the geometric method in Sec. 5-19. This was intended to generate the relation x2 = log™ xi (48) in the domain 1 ^ x\ 2s 10. The design constants established in Sec. 5-19 will be taken as 210 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-8 go m 1.055002 = 1.11303, £i gi = -- = 0.70700, 92 A, = 0.55000, (49) gz = x<0) = 2.61798 (= 150.000°), g4 = fr, = -0.10666 (= -55.00079), Qb = Xi°> = -2.03330 (= -116.500°), g6 = k2 = 1.57079 (= 90.000°). All constants are given to the fifth decimal place, or a thousandth of a degree, since this number of digits will be carried through the further calculations. We have first to choose a suitable spectrum of values for X\. A uniform distribution of values in this spectrum would yield a relatively Table 7-1. — Calculation of 8gis) ' x[s) - x[0) X[s) degrees sin X[s) cos X[s) x(») _ x(o) degrees 0 0.00000 150.000 0 . 50000 -0.86603 0.0 -116.50 1 0 . 25892 148.418 0 . 52371 -0.85189 0.1 - 107 . 50 2 0 . 58489 146.426 0 . 55306 -0.83317 0.2 - 98.50 3 0.99526 143.917 0.58901 -0.80817 0.3 - 89.50 4 1.51189 140.761 0.63256 -0.77451 0.4 - 80.50 5 2.16228 136.786 0.68472 -0.72880 0.5 - 71.50 6 2.98107 131.782 0.74568 -0.66630 0.6 - 62.50 7 4.01187 125.483 0.81428 -0.58046 0.7 - 53.50 8 5 . 30957 117.552 0 . 88659 -0.46255 0.8 - 44.50 9 6.94328 107.569 0.95336 -0.30185 0.9 - 35.50 10 9.00000 95.000 0.99619 -0.08716 1.0 - 26.50 • sin X[s) cos X[s) yw [/(«) VoS) 0 -0.89493 -0.44620 0.84571 0.63313 1.11608 -0.00303 1 -0.95372 -0.30071 0 . 89481 0.56310 1.11777 -0.00474 2 -0.98902 -0.14781 0.93497 0.49224 1.11647 -0.00344 3 -0.99996 0.00873 0.96641 0.42382 1.11357 -0.00054 4 -0.98629 0.16505 0.98968 0.36164 1.11025 0.00278 5 -0.94832 0.31730 1.00567 0.31022 1 . 10761 0.00542 6 -0.88701 0.46175 1.01505 0.27496 1 . 10593 0.00710 7 -0.80386 0 . 59482 1.01782 0.26246 1 . 10484 0.00819 8 -0.70091 0.71325 1.01232 0.28069 1 . 10358 0.00945 9 -0.58070 0.81412 0.99341 0.33883 1.10167 0.01136 10 -0.44620 0.89493 0.94972 0.44617 1.10104 0.01199 Sec. 7-8] GAUGING A THREE-BAR LINKAGE 211 poor check in the range of small xh where fractional errors tend to be greatest. It is better to choose a uniform distribution of spectral values for x2 ; we shall take 4*> x{8) 0.1s H)o.i. 0, 1, 2, 10. (50) The calculation of the gauging error of this linkage is shown in Table 7-1. The gauging parameter g^ is constant to within one per cent; the required length of the connecting bar, V^o, is constant to within one- ■0.010 0.005 Fig. 7-3. — Gauging error in the first logarithmic linkage. Solid line, result of direct calculation. Dashed line, an approximation with slowly varying curvature. half per cent. Figure 7-3 shows the gauging error £<7(0S) plotted against the homogeneous input variable h[s). The gauging error is not large, but it is evident that it can be made much smaller; this linkage has only one point of precision, whereas it should be possible to obtain five (Sec. 7-3). We can proceed in the following way to make a reasonable choice of the points which are to be established as points of precision. Through the curve of dg{0s) is drawn the dashed line of Fig. 7-3, which follows it closely but with minimum variation in curvature, intersecting it in five points. If we establish these points as points of precision, we will 212 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-8 be making a change in 5g08) which also has slowly varying curvature, and which must therefore approximate closely to the dashed curve; the residual gauging error should then be nearly equal to the vertical separa- tion of the two curves in Fig. 7-3. For convenience, let us choose instead to establish points of precision at s = 3, 6, 9, and 10. The fifth point should lie between s = 0 and s = 1, and it would not be entirely satis- factory to take either of these as points of precision. Instead of taking s = 0.5 as the fifth point, we can obtain the same result by requiring that s = 0 shall be, not a point of precision, but a point where there is a predetermined error: 5g0(0) = 0.0019, as read from Fig. 7-3. That is, instead of solving Eqs. (18) or (47) with 5g00) = -0.0031, which would make s = 0 a point of precision, we shall solve them with fy(0) = -0.0050. We shall choose to solve Eq. (47). On calculation of the Q's by Eqs. (39) to (45), these equations take on the following form, for s = 0, 3, 6,9, 10, respectively: -1.00000 Ag0 - 0.12545 Aq1 - 1.03935 Ag2 - 1.04898 Ag3 + 0.00000 Ag4 + 0.18925 Ag5 + 0.00000 Ag6 = -0.00500, -1.00000 Aq0 + 0.22671 Agx - 0.96267 Aq2 - 1.03066 Ag3 - 1.02577 Ag4 + 0.43224 Ag5 + 0.12967 Ag6 = -0.00054, - 1.00000 Ago + 0.57370 Aql - 0.77340 Aq2 - 0.88136 Ag3 - 2.62740 Ag4 + 0.71259 Ag5 + 0.42755 AqQ = 0.00710, - 1.00000 Ago + 0.84480 Aq1 - 0.30102 Aq2 - 0.62289 Aq3 (51) - 4.32490 AqA + 1.00551 Ag5 + 0.90496 Aq6 = 0.01136, -1.00000 Ago + 0.90721 Ag: - 0.02447 Ag2 - 0.52725 Ag3 - 4.74525 Ag4 + 1.04901 Ag5 + 1.04901 Ag6 = 0.01199. Since there are here two fewer equations than there are variables, it is possible to fix two of the variables arbitrarily, subject only to the condi- tion that all Ag's shall be small. On eliminating Ag0, Agi, Ag2, and Ag3 from these equations we obtain 0.00294 Ag4 - 0.05528 Ag5 - 0.05390 Ag6 = 0.00229. (52) The coefficient of Ag4 is small; Ag4 can be chosen arbitrarily with little effect on the relation between Ags and Ag6. It is therefore reasonable to set Ag4 = 0. (53) We can then solve Eqs. (51) for each of the Ag's in terms of Ag6, finding, for instance, Ag3 = 0.05486 - 0.85468 Ag6, (54) Ag6 = -0.04142 - 0.97503 Ag6. Sec. 7-8] GAUGING A THREE-BAR LINKAGE 213 If Ag3 and Ag5 are to be small, we must keep Ag6 small, since its coef- ficients are large. The best value of Ag6 is approximately zero; a posi- tive value will increase the magnitude of Ag5, a negative value that of Ag3. We shall therefore choose Ag6 = 0, (55) and find in consequence (56) Ago = -0.04512, \ Agi = 0.04272, / Ag2 = -0.01985, \ Ag3 = 0.05486, \ Ag5 = -0.04142. / By Eqs. (46), Ago = Ago = -0.04512, Agi = i Agi = 0.02136, Ag2 = - i Ag2 = 0.00992, A^3 = ^- Ag3 = 0.03879, *Qi AQi = + 2g"i Aqi = °' Ag5 = - ^- Ag5 = 0.03765, 472 Ag°= ~iAg6 = a Finally, by Eq. (12), g0 = 1.06791, g[ = 0.72836, g'2 = 0.55992, g'z = 2.65677 (= 152.222°) , g[ = -0.10666, (57) (58) g[ = -1.99565 (= -114.342°), g'6 = 1.57079. To check the performance of the linkage with the new constants g'i, we compute the new gauging error 5g0(s). This is shown in Table 7-2, together with the values 6 Wo(8)U = W - J Q?> Ag; (59) ;=o predicted by a theory in which only first-order terms in the Sg's are retained. The difference between these two quantities, denoted by yis), represents the neglected quadratic and higher terms [Eq. (20) J. Figure 7-4 shows these quantities graphically, 7(s) appearing as the vertical separation of the full and dashed lines. 214 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-8 Table 7-2. — Calculation of dg'0 •) s Z« sinX^ cos X[a) xp sin X<*> cos XP 0 152.222 0.46605 -0.88476 -114.342 -0.91110 -0.41218 1 150.640 0.49030 -0.87156 -105.342 -0.96436 -0.26458 2 148.648 0.52029 -0.85399 - 96.342 -0.99388 -0.11046 3 146.139 0.55718 -0.83039 - 87.342 -0.99892 0.04637 4 142.983 0.60205 -0.79846 - 78.342 -0.97937 0.20207 5 139.008 0.65595 -0.75480 - 69.342 -0.93570 0.35279 6 134.004 0.71929 -0.69471 - 60.342 -0.86899 0.49482 7 127.705 0.79117 -0.61160 - 51.342 -0.78089 0 . 62467 8 119.774 0.86799 -0.49658 - 42.342 -0.67355 0.73914 9 109.791 0.94093 -0.33859 - 33.342 -0.54964 0.83540 10 97.222 0.99207 -0.12571 - 24.342 -0.41218 0.91110 - yw [/(.) 9o{8) ^ (S<#8)) exp. yto 0 0.84960 0.58636 1.06564 0.00227 0.00197 0.00030 1 0.89708 0.51333 1.06826 -0.00035 -0.00059 0.00024 2 0.93545 0.43984 1.06852 -0.00061 -0.00041 -0.00020 3 0.96514 0.36921 1.06781 0.00010 0.00000 0.00010 4 0.98688 0.30529 1.06713 0.00078 0.00083 -0.00005 5 1.00168 0.25270 1.06722 0.00069 0.00083 -0.00014 6 1.01047 0.21694 1.06811 -0.00020 0.00000 -0.00020 7 1.01349 0.20477 1.06909 -0.00118 -0.00101 -0.00017 8 1.00934 0.22445 1.06915 -0.00124 -0.00118 -0.00006 9 0.99309 0.28563 1.06781 0.00010 0.00000 0.00010 10 0.95337 0.39829 1.06755 0.00036 0.00000 0.00036 0.005 0.005 Fig. 7-4. — Gauging error in the first improved logarithmic linkage. Full line, result of exact computation. Dashed line, values expected on linear theory. Sec. 7-8] GAUGING A THREE-BAR LINKAGE 215 We have thus established precision points at positions shifted only slightly from those initially — and rather arbitrarily — selected. The result of this first calculation might very well be accepted as final. It is, on the other hand, easy enough to make a first-order correction for the effects of the quadratic and higher terms. We have only to replace 8g(08) on the right-hand side of Eq. (51) by 7(s), and solve for new Ag's and Ag's to be added to those already obtained. As before, we choose arbitrarily A#4 = Aq& = 0. The second-order corrections to the gr's are then found to be A20O = 0.00246, Afy 0.00170, A2g2 = -0.00130, A203 = -0.00292, A2#4 = 0.00000, tfgh = -0.00328, A2gr6 = 0.00000. The new and final constants of the linkage are (60) g'l ifi g'l = 1.07037, = 0.72666, = 0.55862, = 2.65385 (= 152.054°), = -0.10666, = -1.99893 (= 114.530°), = 1.57079. (61) The final values of the gauging parameter and the gauging error are shown in Table 7-3, together with the resulting error in the homogeneous Table 7-3. — Characteristics of the Second Improved Logarithmic Linkage C* &C8) 1 5XJ'(S\ radians degrees 8h2 3 292Qis) 0 1.06846 0.00191 -4.7295 -0.00903 -0.517 -0.00574 1 1.07105 -0.00068 -3.3403 0.00227 0.130 0 . 00144 2 1.07123 -0.00086 -2.5673 0.00221 0.127 0.00141 3 1.07039 -0.00002 -2.0707 0.00004 0.002 0.00002 4 1.06958 0.00079 -1.7212 -0.00136 -0.078 -0.00087 5 1.06956 0.00081 -1.4594 -0.00118 -0.068 -0.00076 6 1.07035 0.00002 -1.2561 -0.00003 -0.002 -0.00002 7 1.07132 -0.00095 -1.0963 0.00104 0.060 0.00067 8 1.07149 -0.00112 -0.9742 0.00109 0.062 0.00069 9 1.07036 0.00001 -0.8902 -0.00001 0.000 0.00000 10 1.07042 -0.00005 -0.8532 0.00004 0.002 0.00002 216 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-8 output parameter, 5h2.^ To compute this we note that sxp whence • « (MIX1 „ _ W1 _ *W « «*•, (62) (63) the conversion to terms of the homogeneous output variable is obvious. These results are also presented graphically in Fig. 7-5. +0.005 1 1+5 -0.005 Fig. 7-5. — Characteristics of the second improved logarithmic linkage. The dashed line gives the form of a correction to be discussed in Sec. 7-9. Xi10)=97.054° = -0.00595 VXf = -115.067° Fig. 7-6. — Second improved logarithmic linkage. The linkage itself is outlined in Fig. 7-6. The constants g0, gi, and g2 determine the lengths of the linkage arms, whereas g* and gr4 determine the nature of the input scale. The linkage is shown with the input arm at either end of this scale — that is, for X\ = 1 and Xi = 10. The output arm is shown in the corresponding positions required by the geometry of the linkage. Because of the structural error in the design, these posi- Sec. 7-9] THE ECCENTRIC LINKAGE 217 tions do not coincide with the ends of the x2 scale determined by g5 and ^e ; the scale readings are those shown in Fig. 7-6. (These have been determined by exact computation; hence one finds x(20) = h20) = —0.00595 in Fig. 7-6, in contrast to the approximate value, —0.00574, in Table 7-3.) 7'9. The Eccentric Linkage as a Corrective Device. — When specified tolerances are very close it may not be possible to meet them by any choice of the parameters of such simple linkages as the three-bar linkage. Reduction of the structural error to tolerable limits then requires intro- duction of new adjustable parameters into the linkage. In many cases one can introduce small additional corrections by a superficial change «.«•-*» <1(xlw-?) Fig. 7*7. — Three-bar linkage modified by a double eccentric linkage. in the structure of the linkage. Replacement of an ideal harmonic transformer by a nonideal one is such a change ; another is the introduction of eccentric linkages at the joints of three-bar linkages or harmonic transformers. These modifications in the structure are mechanically sound, and permit one to make use of all previous computations — an important economy in effort. Figure 7-7 shows a three-bar linkage modified by the introduction of an eccentric linkage at either end of the connecting bar. The moving pivots of the input and output cranks carry planetary gears meshing with stationary gears. The connecting bar is not pivoted to the cranks, but to the eccentric pivots Ex and E2 on the planetary gears; the ends of the connecting link do not move with the pivots of the cranks, but about them in circles with radii ei and e2 — usually small. Thus the distance between the ends of the cranks is not a constant; in effect, go can be made 218 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-9 to vary, as is required for more precise generation of the given functional relation. Each eccentric linkage may be characterized by three constants: the tooth-ratio t of the stationary to the planetary gear, the eccentricity e of the planetary gear, and the angular position of the crank (denoted by Xi = a/t) for which the eccentric pivot lies on the center line of the crank, at a maximum distance from the frame pivot. The double eccentric linkage in Fig. 7-7 thus provides the designer with- six additional parameters to adjust. Obviously, for greatest precision one should adjust all constants of the device simultaneously. Usually one can obtain very satisfactory results by accepting as fixed all constants determined in previous design work, varying only the constants of the eccentric linkages, Indeed it is often satisfactory to use only a single eccentric linkage, with consequent reduction to three in the number of constants to be adjusted. To determine the constants of the eccentric linkage one can employ the gauging-parameter method in a somewhat modified form, valid so long as the eccentricity of the linkage is small. In dealing with a modified three-bar linkage one can advantageously use the squared length of the connecting link, g0, as the gauging param- eter. Reference to Fig. 7-7 shows that introduction of the first eccentric linkage has the same effect on g0 as a change 8g[s) in the length of the input crank and a change 8X(f in its angular position, where 8g[* = ex cos (hXp - ai), (64a) dX\* = (-) sin (tiXP - a,), (646) to terms of the first order in the small quantity ei. Similarly, introduc- tion of the second eccentric linkage has the effect of changing g2 and 5X2 by, respectively, bgf = e2 cos (kX^ - a2), (65a) = I - ) SI] w sin (*2X2S) - a2). (656) The resulting change in the gauging parameter is /W<0 ,W8> /W*> AY'-8'* Ag° dgx 9l + dX? * ^ dg2 d92 + dX? 2 ' (bbj or, by Eq. (38), A<7(08) = W + QPIXP + &PW + G^hXf. (67) It is thus the sum of four sinusoids multiplied by the slowly varying G's. Combining Eqs. (64), (65), and (67), one can write Sec. 7-9] + e2 THE ECCENTRIC LINKAGE <w>* + (fe)' /Gis)\' sin sin hXf - ax + tan-' (^) feX?' - <*2 + tan-' (^) 219 (68) Here the contribution of each eccentric linkage to the gauging parameter is expressed as a sinusoid with adjustable frequency, amplitude, and phase constant, the second and third of these quantities being subject to slow variations of predetermined character. The difference in effect of eccentric linkages on the input and output cranks arises partly from differences in these variations, but principally from the fact that the argument of the sinusoid is in the first case a linear function of X\, in the second case a linear function of X2. It is possible to use the additional flexibility provided by eccentric linkages to increase the number of precision points, if all constants of the device are adjusted simultaneously. When only the constants of the eccentric linkages are to be adjusted it is usually desirable to leave undisturbed the precision points already established. One can make Ag(0s) vanish at five previously established precision points by adjustment of the five constants h, t2, ah a2, and e2/e\. Then Ag(0s) will have the same zeros as the gauging error of the original three-bar linkage, and usually the same general form; by appropriate choice of the remaining constant, say ei, one can give it roughly the same magnitude. The completed linkage will then have the same precision points as before, but smaller gauging errors. When a single eccentric linkage is to be used one can leave undisturbed only two precision points. Example. — As an example, we shall further reduce the structural error of the logarithmic linkage of Fig. 7*6, using a single eccentric link- age. The design procedure is then extremely simple, but requires the exercise of some judgment if best results are to be obtained. The error function of the original linkage, as shown in Fig. 7-6, has a generally sinusoidal character. The points of precision occur for h2s) = 0.05, 0.3, 0.6, 0.9, 1.0, h[s) = 0.0125, 0.1125, 0.3325, 0.772, 1.0. (69) Except for the last, they are quite evenly spaced in X2, but unevenly spaced in X\\ they have about the same distribution as the nulls in a sinusoid with argument linear in X2. If a single eccentric linkage is to be used it should be placed on the output crank; it will then be possible to leave four, and not just two, of the points of precision essentially unchanged. We will have then, on introducing the Q's in place of the G's, A<7(o8) = 2e2[(Q</>)2 + (Q(58))2F sin t2X? - a2 + tan sin i -(!)]■ (70) 220 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-9 The nulls of this expression occur when CD - - ttKW — a2 + tan" 180°. (71) Table 7-4 shows the values of Q28)/Qb8) at the previously established precision points, s = 0.5, 3, 6, 9, 10, and the values of hX^ — a2 required if these points are to be nulls of Ag(Qs); n has been assigned the values 0, 1, 2, 3, 4 at the successive nulls. ^K).00l 1 o £-0.001 ,6h2 Dr^ J 0.5 _ — -" 1.0 *< ^ r— V ~K Fig. 7-8. — Structural error in the final logarithmic linkage. Table 7-4. — Values of (t2X[s) — «2) Required for Vanishing of Ag(0s) s Qi8)/Q[3) ttX™ - a2, degrees 0.5 3.0 6.0 9.0 10.0 -4.6543 -2.2272 -1.0853 -0.2994 -0.0233 77.9 245.8 407.3 556.7 721.3 Let us choose to retain the points s = 0.5 and s = 9 as points of preci- sion. Taking the values of X(2S) in Table 7-1 as sufficiently accurate, we then require fe(-110°) - a2 = 77.9°, U (-33.5°) - a2 - 556.7°, (72) whence U = 6.26, a2 = -766.5°, a2 (t3) = -122.4°. Sec. 7-9] THE ECCENTRIC LINKAGE Xx(0) =152.054* 221 0.55862 e2= 0.00041 (exaggerated) Fig. 7-9. — Final logarithmic linkage. The corresponding values of Ag(os)/e2 are shown in Table 7-5, and are plotted (dashed curve) in Fig. 7-5. This curve has roughly the same form as the residual error 8g"{s) of the linkage which is to be improved ; inspection will show the Ag(08) gives about the best approximation to 8g'0'w when e2 = 0.00041. (74) Table 7-5. — Calculation of Structural Error of Final Logarithmic Linkage 5 e2 &A* *C(S) 5h,2 0 1 2 3 4 5 6 7 8 9 10 1.070 -1.066 -2.113 -0.975 1.1845 2.0909 0.7207 -1.4760 -1.9617 -0.0107 1.9882 0.00042 -0.00042 -0.00084 -0.00039 0.00047 0.00084 0.00029 -0.00059 -0.00078 -0.00004 0.00079 0.00149 -0.00026 -0.00002 0.00037 0.00032 -0.00003 -0.00027 -0.00036 -0.00034 0.00005 -0.00084 -0.00420 0.00050 0.00003 -0.00044 -0.00032 0.00002 0.00019 0.00023 0.00019 -0.00003 0.00041 222 FINAL ADJUSTMENT OF LINKAGE CONSTANTS [Sec. 7-9 The resulting values of Ag(08) and of the final gauging error *0o"(-) = W8) ~ A<tf> (75) are given in Table 7-5, together with the resulting error in the homo- geneous output variable; the last two quantities are plotted in Fig. 7-8. The structural error remains less than 0.05 per cent except in the immedi- ate neighborhood of hi = /t2 = 0, where it abruptly rises to 0.4 per cent. The linkage is outlined in Fig. 7-9 in its configuration for s = 2, very near to one of its precision points. CHAPTER 8 LINKAGES WITH TWO DEGREES OF FREEDOM Functions of two independent variables are usually mechanized by three-dimensional cams (Fig. 1*24), which are expensive to manufacture, and rather bulky ; they are, however, easy to design and have very wide application. Bar linkages with two degrees of freedom can also serve to mechanize functions of two independent variables. These linkages have the advantages of being flat and small, of giving smooth frictionless performance allowing appreciable feedback, and of being relatively inexpensive to manufacture in quantities. They are, on the other hand, relatively difficult to design, having always residual structural errors which must be brought within the specified tolerances. The mathe- matical design of these linkages will be treated in the remainder of this book. Basic concepts needed by the designer will be introduced in the present chapter. Succeeding chapters will show, partly by precept and partly by example, how to design linkage multipliers or dividers (Chap. 9) and linkage generators of more general functions of two independent variables (Chap. 10). 8*1. Analysis of the Design Problem. — Mechanisms with two degrees of freedom have at least one output parameter Xk functionally related to two input parameters Xi and Xj\ Xk = F(Xi} X,). (1) If the domain of definition D of this relation is a rectangle, Xim ^ Xi ^ XiM, Xjm ^ Xj ^ XjM, (2) the mechanism is said to be "regular." To such a mechanism one may add functional scales that establish relations between the parameters Xi, Xj, Xk, and corresponding variables Xi, Xj, Xk, respectively. The mechanism will then serve to establish a functional relation xk = f(xif Xj) (3) between these variables; we may say that the device, mechanism plus scales, mechanizes Eq. (3). If this relation of the variables is to be single- valued, it is necessary that to definite values of the input variables there correspond definite values of the input parameters, and that to a definite value of the output parameter there corresponds a definite value 223 224 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 81 of the output variable, form, The scales must then establish relations of the Xi = (Xt\xi) • xi} Xj = \Xj\Xj) - Xj, Xk = (xk\Xk) ' Xk, (4) where all three operators (but not necessarily their inverse operators) are single- valued. If Eqs. (4) are of linear form, Xi = X<°> + Uxi - x^), X, = X}» + hi?* ~ *}0)), Xk = x<g> + Kk(Xk - Xk0)), (5) the device provides a " linear mechanization" of Eq. (3). When a mechanism is to be a component of a more complex computer, it is often, but not always, required to provide a linear mechanization of the relation between input and output variables. Any mechanism generating a function F of two independent param- eters may be represented schematically as in Fig. 8*1. This representation is sufficient in the case of three-dimensional cams, which can generate in one step, so to speak, any well-behaved function of two independent parameters. Simple bar linkages, on the other hand, can generate only a restricted class of functions; to mechanize a given rela- tion between parameters one must usually build up a more complicated structure, a combination of one or more simple linkages of two degrees of freedom and several link- ages of one degree of freedom. It is then necessary to consider the internal structure of the function generator F. Let G denote a simple bar linkage with two degrees of freedom, gen- erating a function of two independent parameters, Fig. 8-1. — Schematic repre- sentation of mechanism gener- ating a function Xk of two independent parameters, Xi and Xj. Y„ = G(Yh Y,), (6) of a restricted class. By combining such a linkage with three linkages having one degree of freedom, as shown schematically in Fig. 8-2, one can generate relations of a much wider class between parameters X-i, Xj, Xk. A more elaborate structure is that shown in Fig. 8-3, which consists of four linkages, each with two degrees of freedom, so connected as to make use of feedback. Theoretically, such structures make possible a further extension of the field of mechanizable functions. In practice it is usually sufficient to use the simpler structure of Fig. 8-2, to which we shall henceforth confine our attention. Sec. 8-1] ANALYSIS OF THE DESIGN PROBLEM 225 We have then to consider structures consisting of a linkage with two degrees of freedom, which establishes a relation [Eq. (6)] between internal parameters Yi} F„ Yk, and three linkages of one degree of freedom, which relate the internal parameters to the corresponding external parameters Xi, Xj, Xk: Yi = (Y,\Xi) ■ Xi, Yt = (r„|Zy) • xh- Xk = (Xk\Yh) ■ Yk. (7) Together, these establish a relation between the external parameters [Eq. (1)]; the functional scales, in turn, convert this into a relation [Eq. (3)] between variables Xi, as,-, xk, which is to be made to approximate as closely as possible to some given relation, throughout a specified domain. The linkage G, with two degrees of freedom, we shall call the "grid generator," for reasons which will become evident later. The linkages Fig. 8-2. — Combination of grid generator and transformer linkages. Fig. -Feedback linkage degrees of freedom. with two Tif Tj, Tk, we shall call the "transformers," because they transform the internal parameters Y into the external parameters X. The division of a mechanism into a grid generator and transformers is to some extent arbitrary; the breakdown of a given functional relation [Eq. (1)] into a grid-generator relation [Eq. (6)] and transformer relations [Eq. (7)] is completely arbitrary. We shall therefore make use of the generalized term "grid generator for a given function" as denoting any linkage with two degrees of freedom which will serve as the linkage G in a mechaniza- tion of the given function. Transformer linkages increase the field of linearly mechanizable func- tions, but not the field of functions mechanizable in the more general sense. A relation xk = /(x», x,) mechanized by a grid generator [Eq. (6)], transformer linkages [Eq. (7)], and functional scales [Eq. (4)] can be mechanized also by associating the same grid generator directly with scales which establish relations Y{ = (Yi\Xi) xk = (xk[Xk) (Xi\xi) • x i= <f>i(xi), (XjIxj) • Xi = 4>i(xj), (Xk\Yk) • Yk = <j>k(Yk). (8) 226 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-2 Transformer linkages in a design thus serve only to change the form of the functional scales — usually to make them linear. It is obvious that the choice of a grid generator is the central problem in the design of a linkage with two degrees of freedom. When a linear mechanization is desired, one can then proceed to design the transformer linkages by methods discussed in the preceding chapters; concerning this latter stage of the work, which offers no new theoretical problems, little more need be said. It is evident that a very simple grid generator may serve if the transformers are made sufficiently complex, whereas another choice of grid generator may make unnecessary the use of one or more transformers. It is important that the transformers not add too much to the complexity of the design; a good grid generator should be simple in structure, and also adapted to use with simple transformer linkages. For instance, we shall see that the common differential is a theoretically adequate grid generator for an important class of functions; its general use in linearly mechanizing these functions is, however, not to be recommended, since the required transformers tend to be excessively complex. In practice one has available a relatively small number of types of linkage suitable for use as grid generators; the available grid-generator functions G belong to several restricted classes. Usually these will not include an exact grid-generator function for the given function; a struc- tural error must be admitted in designing the grid generator. Structural errors must also be admitted in the design of the transformer linkages. Thus it is always important in designing such mechanisms to make a final adjustment of all available constants, in order to minimize the over-all structural error. In summary, mechanization of a given function of two independent variables involves the following steps: 1. Choice of a suitable type of grid generator. 2. Selection of the constants of the grid generator. 3. Design of the transformer linkages. 4. Final adjustment of all constants of the mechanism. The ideas to be developed in the remainder of this chapter are essential for the first of these steps; they also form a foundation for the procedures required in the second step, which will be described in later chapters. 8-2. Possible Grid Generators for a Given Function. — It is very easy to give a formal characterization of all functional relations which can be mechanized by use of a given grid generator. Combining Eqs. (6) and (8), we see that these are the relations which can be expressed as Xk - f(xi, xt) = **{£[&(*<), +K*i)]}, W where G is the given grid-generator function and <f>if <j>j, fa are arbitrary Sec. 8-2] GRID GENERATORS FOR A GIVEN FUNCTION 227 single-valued functions of their arguments. Conversely, to mechanize a given functional relation xk = fixi, xj) (3) one can employ a grid generator with parameters related by Yk = G(Yh y,) = flr'M-fcrW, #*&,)]}, (io) where 4>Tl, 4>Tl, &I1 are the inverse of arbitrary single-valued functions <f>i, fay fa- The relations expressed in Eqs. (9) and (10) can also be expressed in terms of contour lines of the functions / and G. Let us plot contours of constant Yk = G(Yi} Yj) in the (Yi, F;)-plane and label them with the cor- v(-2) y(0) H jrH) h yd) y(2) yt|-2)y(-i)r(0)y(i) ym (a) Fig. 8-4. HYtxy) (Ok r=<t>kwr) Yi -Topological transformation of contours. responding values of Yk (Fig. 8-4a). Next let us introduce a change in the independent variables, defined by the equations Yi = frfa), Yj = <fo(av), (11) where <fo and <f>}- are single-valued functions of these arguments. Replot- ting the contours of constant G in the (xi} Xj) -plane (Fig. 8-46), we obtain lines of constant f(xi, xj), as defined by Eq. (9). If these contours are relabeled with values of xk given by 4r) - MY?), (12) they will represent the functional relation Xk = f(xi} Xj) (13) defined by Eq. (9), for a particular choice of the functions fa, fa, fa. It is thus clear that a given grid generator can be used in mechanizing a given function if the contours of constant G(Yi, Fy) can be transformed into those of constant f(xi} xj), or conversely, by any topological trans- 228 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-3 formation of the form of Eq. (11), with relabeling of the contours accord- ing to Eq. (12). Formal relations such as Eqs. (9) and (10) are not of great value in practical design work. The graphic presentation of these relations by means of systems of contour lines is of more interest, but as an indication of a direction of development, rather than as a completed idea. What is really needed is a means of characterizing given functions, on the one hand, and available grid generators, on the other, which will make it clear at once whether or not a given grid generator can be used in mechanizing a given function. Even more valuable will be a means of characterizing a given function which will assist one in designing a new and satisfactory grid generator. In both respects the idea of "grid structure of a func- tion" is of fundamental importance. 8-3. The Concept of Grid Structure. — The representation of a function of two independent variables by a grid structure is an extension of the familiar representation by a set of contours of constant value of the dependent variable. It will here be introduced in a specialized form, satisfactory for the classification of functions; in later sections it will be generalized and applied in design work. Rectangular Grid Structure with Respect to a Center S and a Contour C. — We have now to construct the grid structure of a functional relation Xk = f(xi} Xj) (14) denned through a domain D in the (xi} a:,) -plane. Let S be a point in the domain D, associated with values of the varia- bles which will be denoted by Xi0), z}0), xp; this is to serve as the " center" of the grid structure. Through S construct the contour B of constant Xk, 40) = K*i, x,). (15) (See Fig. 8-5.) Next choose an adjacent contour C, denned by xk = xjr«. (16) This, together with the point S, will fix the grid structure that is to be constructed. Through S construct the vertical line Xi = xl0), intersecting the con- tour C at the point (x[0), zj-1), xi~v). Through this latter point construct the horizontal line x; = x)~l), intersecting the contour B at the point (x(il), a:j_1), x(k0)). Through this point, in turn, construct the vertical line Xi = xl1*, intersecting the contour C at the point (x^1}, z}-2), x(£~l)). Continuation of this process extends the steplike structure of lines between the two contours, both above and below S, and defines sequences of values of the two independent variables: Sec. 8-3] THE CONCEPT OF GRID STRUCTURE 229 (—2) „(— 1) ~(0) r(l) r(2) . . . , Xt- , Xj ) ^i } "^i ) ^i ) r(-2> r(-D ~{p) xa) xw The rectangular grid of lines Xi = x (r) and xv (17) (18) will cover part, but not always all, of the domain D. This rectangular grid serves to define a system of contours xk = sjp, (19) which, together with this grid itself, will make up the " rectangular grid structure of the function, defined with respect to the center £ and the contour C." Xk — xk , Fig. 8-5. — Ideal grid structure. Ideal Grid Structure. — The rectangular grid has been so constructed, and its lines so numbered, that a single contour, (20a) passes through all grid intersections for which r + s = 0, (206) and a single contour, xk - a4-», (21a) passes through all grid intersections for which r + s = -1. (216) 230 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-3 There is an important class of functions such that, no matter how the center S and the contour C are chosen, there will be a single contour, xk = 4°, (19) passing through all grid intersections for which r + s = t, (22) t being any integer, positive or negative. Such a function will be said to have " ideal grid structure." An ideal grid structure (denned with respect to a center S and a contour C) will consist of the rectangular grid specified above, plus all the contours of constant Xk which pass through the intersections of the grid. Such a grid structure will appear as shown in Fig. 8-5. This grid struc- ture can also be described as consisting of three families of curves, given by Eqs. (17), (18), and (19), such that through every point of intersection there passes a curve of each family. This description will re- main valid even when the con- cept of ideal grid structure is generalized. N onideal Grid Structure. — When different contours of con- stant Xk pass through grid inter- sections characterized by the same value of (r -f- s), the grid struc- ture will be said to be "nonideal." Figure 8-6 represents an extreme case of nonideal grid structure. When the grid structure is non- ideal one cannot distinguish by the single index (r + s) the contours of the family defined by the grid; one might instead label each curve with the two indices, r and s, of the corresponding grid intersection, as shown in Fig. 8-6. It is not convenient to consider all these contours as belonging to the grid structure of the function, nor would this contribute to the clarity with which the grid structure represents the properties of the function. It is sufficient to include only one such contour for each value of (r + s), label- ing it with this quantity as the single index. The choice of the contours to be included is to some degree arbitrary. We shall consider a nonideal grid structure to consist of the rectangular grid defined in the usual way, plus the contours of constant xk that pass through the grid intersections with r = s — n, plus intermediate contours that interpolate smoothly Fig. 8-6. — Nonideal grid structure. Sec. 8-3] THE CONCEPT OF GRID STRUCTURE 231 between these and therefore pass near the intersections with r = s + 1 = n and r + 1 = s = n. (A precise method for choosing these intermediate contours will be indicated in Sec. 8-6.) (-5) ,H) „(-3) (-2) (-1) (0> „(1) r(2) (3) „(4) „(5) ** xi xi xi xi xi xi xi xi Fig. 8-7. — Grid structure of the function Xk = — 7= (xi2 + XiXj + xy2)a V3 Figure 8-7 illustrates a typical nonideal grid structure, that of the function xk « -~= (sj + x#i + xf)». (23) The point ar< = Xj = £& = 1 has been chosen as the center S, and the contour C is that for which Xk = 0.9 = xljrl). The contours are sym- metrical with respect to the dotted line in the figure, and so is the rec- 232 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 84 tangular grid. It will be observed that near the contours B and C, and near the line r = s, the contours pass very nearly through the grid inter- sections. Away from these lines the nonideal character of the grid struc- ture becomes increasingly apparent, as the contours pass farther and farther from the grid intersections. Grid Structure in the Neighborhood of a Center. — The greater the distance from the center S to the adjacent contour C of the grid struc- ture, the more coarsely does the grid structure represent the properties of the function. In order to define for a function "the grid structure in the neighborhood of a center," one must allow the contour C to approach the center S, and concentrate attention on a very small region about the center which, nevertheless, contains a considerable number of grid lines. One can expand in Taylor's series about the center S any well-be- haved function: * - /(*<, *,) = * + (jQ, (x, - *<»>) + (g)s fe. - *;«>) + terms of higher order in (xi — x[0)) and (x3- — x$0)). (24) In the immediate neighborhood of S the quadratic and higher terms in Eq. (24) can be neglected. To this approximation the contours of constant xk are parallel straight lines, the grid consists of identical rectangles, and the grid structure is ideal. Thus one can say that the grid structure of any well-behaved function is ideal in the neighborhood of its center. The practical significance of this statement, which will be brought out more completely in later sections, is this: It is always easy to find a grid generator for a function if the domain of mechanization is sufficiently restricted; what is difficult is to find grid generators useful throughout extended domains. 8-4. Topological Transformation of Grid Structures. — It has been shown in Sec. 8-2 that the topological transformation & = *,-(*,•), (11) carries contours of the function Yk = G(F„ Y,) (6) in the (F», F7)-plane into contours of the function %k = f(xi} X,) (13) in the (xi, z;)-plane. This transformation carries vertical straight lines in the (Ft, F;)-plane into vertical straight lines in the (xi} £/)-plane, and horizontal straight lines into horizontal straight lines. Indeed, the reader will easily see that the idea of grid structure has been so defined Sec. 8-5] THE SIGNIFICANCE OF IDEAL GRID STRUCTURE 233 that if this transformation carries a center Sy in the (Yi, F,)-plane into a center Sx in the (afc, x,)-plane, and a contour CY into a contour Cx, then it carries the complete grid structure of the function G(Yi} Yj), defined with respect to SY and CY, into the grid structure of the function f(xi, xi), defined with respect to Sx and Cx. The values of the variables associated with the grid lines and contours will be transformed according to Eqs. (11) and (12), but the indices r, s, t, will be unchanged. The main conclusion of Sec. 8-2 can therefore be restated in the follow- ing terms : A given grid generator can be used in the exact mechanization of a given function if, and only if, there exists a topological transforma- tion, of the form of Eq. (11), that carries each grid structure of the function G(Yi, Yj) into a corresponding grid structure of the given func- tion/^, Xj). In practice, of course, all that need be shown is that some grid structure of the function G(Yi} Yj), with sufficiently small meshes, can be thus transformed into a corresponding grid structure of the function f(xi} xi), with errors within specified tolerances. The topological transformation cannot change intersection properties of the lines of the grid structure; it must then transform an ideal grid structure into another ideal grid structure, a nonideal grid structure into another nonideal one. It follows that a given function with an ideal grid structure can be mechanized exactly only by a grid generator with ideal grid structure, a given function with nonideal grid structure only by a nonideal grid generator. In Sec. 8-5 it will be shown that all functions with ideal grid structure can be mechanized by any grid generator with ideal grid structure, such as the common differential. In the case of nonideal grid structures the situation is not so simple. There are many diif erent ways in which a grid structure can be nonideal ; it is in general possible to determine whether or not a given grid generator will serve in the mechanization of a given function only by making a detailed comparison of their respective grid-structure properties. In Sec. 8-6 there will be indicated the basic ideas of a systematic method for choosing from among a number of given types of grid generator the one which is most suitable for the mechanization of a given function. Unfor- tunately, this method cannot suffice for the design of nonideal grid gen- erators until an extensive file of grid structures has been accumulated. In the present state of the art it is necessary to design a grid generator ab initio for each given function; the way in which this can be done, by a study of its grid structure, will be indicated in Sec. 8-7, and illustrated at length in Chap. 10. 8-5. The Significance of Ideal Grid Structure. — It will now be shown that if a functional relation xk = f(xif x3) (25) 234 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-5 has ideal grid structure, then there exists a topological transformation Yi = *&), ) (26a) Yi = fcfe), (266) xk = <f>k(Yk), ) (26c) such that Yk = Yi + F,. (27) In other words, if the functional relation Eq. (25) has ideal grid structure it can be expressed as <H-\xk) = tfifa) + *(**). (28) It will follow immediately that this function can be mechanized using a differential as grid generator, together with transformer linkages and scales which establish the relations of Eqs. (26). X to r x (i) x \ V j\ \ i & \ N \1 \ s\'\ -(0) XJ \ \l \ \ \N \ \ v.H) V \ *J \ N \ \ \^\. Xk (0) Fig. 8-8. — Subdivision of an ideal grid structure. Let us consider the ideal grid structure defined with respect to a center S and a contour C, as shown by the solid lines of Fig. 8-8. Asso- ciated with each intersection in this grid structure are values of the indices r, s, and t, such that r + s = L (29) The index r is a single-valued function of the x;-coordinate of the inter- section, s is a single-valued function of Xj, and xk is a single- valued function of t. In short, the indices r, s, t have all the characteristics which should be possessed by the parameters F,, F,, Yk, respectively, except that they are defined only for a discrete sequence of values, instead of as continuous functions of xi} Xj, xk. We shall now show that the definition of the indices can be extended to apply to a continuum of values; the theorem above will then follow on identification of r, s, t with F», F3, Yk, respectively. Let us consider the portion of Fig. 8-8 lying between the contours B and C, and between the lines z{-0) and x^K It is clearly possible to choose a contour C such that a step structure constructed between the contours Sec. 8-5] THE SIGNIFICANCE OF IDEAL GRID STRUCTURE 235 B and C passes from the center S to the point (#p, zj_1), o40)) in two steps, instead of one. Now let us construct a grid structure with respect to the center $ and the contour C (solid and dashed lines of Fig. 8-8). It is clear from the method of construction that this new grid structure includes the contour C as its first contour beyond C". It follows immedi- ately that every fine of the original grid appears in this new one; in addition, there is a new line interpolating between each pair of adjacent lines in the old grid. Instead of reassigning integral indices to all lines of the new grid, we shall retain the old indices for the old lines and assign half -integral indices to the intervening lines. All indices are just half as large as they would have been if the construction had been begun with the center S and the contour C '; Eq. (29) is still satisfied, but half-integral indices may occur in it as well as integral ones. In the same way we can construct a new grid structure in which C is the second contour (rather than the first) beyond S, and can assign to the fines of this structure quarter-integral in- dices which satisfy Eq. (29). Con- tinuing to subdivide the original grid in this way, we can define grid-struc- ture lines corresponding to arbitrary values of r, s, t, in a continuous range, throughout maintaining the validity of Eq. (29) . These indices appear as functions of xi} x3-, xk, having the form of Eqs. (26) ; it is only necessary to identify r, s, t with Yi} Yh Yk, to complete the proof of the theorem. As examples of functional relations with ideal grid structure we may take xk = Xi + xu (30) with grid structure shown in Fig. 8-9, Fig. 8-9. 0 +1 Grid structure of xt- + Xj = xk. x\ = x\ + x% with grid structure shown in Fig. 8-10, and Xi xk = -, or In xk = In xi — In Xj (31) (32) (33) with grid structure shown in Fig. 8-13. An alternative statement of our result is the following: If a functional relation has ideal grid structure, it is always possible to apply a topo- 236 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8 5 logical transformation of the form of Eqs. (26) that will transform this grid structure into the form shown in Fig. 8-9, within some domain of the variables. Possible limitations of the domain of this transformation will be evident on comparison of Figs. 8-9, 8-10, and 8-13. The grid structures of Figs. 8-9 and 8-10 correspond closely in the first quadrant, and the general form of the required transformation of horizontal and vertical rH) r(0) _tt) „(2)„(3) (4) (5) *»• *« *i *» *i xi xt Fig. 8- 10.— Grid structure of x^ + xf = xk2. coordinates is clear enough; on the other hand, it is also clear that a transformation which will serve to carry one grid structure into the other in the first quadrant will not have this effect in the second quadrant, or the fourth. This is due to the fact that the transformation Eq. (26a), as defined by the grid structure, ceases to be single- valued when the contour C is followed through a point of infinite slope; similarly, Eq. (266) ceases to be single-valued when C is followed through a point of zero slope. Together, these limitations restrict the transformation from Fig. 8*9 to Sec. 8-5] THE SIGNIFICANCE OF IDEAL GRID STRUCTURE 237 Fig. 8-10 to corresponding quadrants. A very different example is provided by Eq. (33). The transformation equations Yi = In Xi, Y3 = - In xu (34) Yk = In xk, which transform Fig. 8-13 into Fig. 8-9, transform the first quadrant of Fig. 8- 13 into the whole of Fig. 8-9; other transformations carry each of the other quadrants of Fig. 8-13 into the whole of Fig. 8-9. We have now proved that any function with ideal grid structure can be mechanized using a differential as grid generator. This is by no means necessary, nor is it usually desirable. It is, in fact, possible to use any grid generator with ideal grid structure in mechanizing any given func- tional relation Xk = ffa, xj) with ideal grid structure; the choice should depend on the mechanical desirability of the device as a whole. In order to make contact with the analysis of Sec. 8-1, let us suppose that it is desired to establish between external parameters X%, Xj, Xk} a given relation Xk = F(Xi} Xi) (35) with ideal grid structure ; this is a problem equivalent to that of finding a linear mechanization of a relation of the form of Eq. (35) between variables #,-, Xj, Xk. Let there be given a grid generator with ideal grid structure mechanizing the relation Yk = G(Yi} Yi) (36) between internal parameters Yi, Y3-, Yk. We have seen that this relation can also be mechanized using a differential as grid generator; Eq. (36) is equivalent to Zk = Zi + Zi} (37) Zi = (Zi\Yi) • Yi} ) Zi - (Zi\Yi) • Y3, (38) Yk = (YAZk) • Zk, with the indicated transformer functions all single- valued. Conversely, the given grid generator, Eq. (36), can be used in mechanizing Eq. (37), as indicated in the inner circle of Fig. 8*11; the transformer functions required are the inverse of those in Eq. (38). We know also that the resulting differential can be used in mechanizing Eq. (35), in combination with transformer linkages generating the relations Zi = (Zi\Xi) • xl, ) Zi = (Zi\Xi) • Xi, (39) Xk = (Xk\Zk) ' Zk, I 238 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-6 as shown in the outer circle of Fig. 8-11. It thus becomes obvious that the given grid generator can be used in mechanizing Eq. (35), by com- bining it with transformers mechanizing the operators (F4X-) = (F.IZi) • {Z,\Xt), {Y,\X,) = (Yt\Z,) ■ (Z,\X,), (Xt\Yt) = (Xk\Zk)-(Zk\Yk). (40) These transformers may be simpler in structure than those required with the simpler differential grid generator, and the domain of operation of the complete device may be more extensive. For example, it is cer- tainly possible to build a multiplier with linear scales, using a differential Fig. 8-11.- -Mechanization of a given function with ideal grid structure by a given grid generator with ideal grid structure. as grid generator, and logarithmic transformers [Eq. (34)] such as that illustrated in Fig. 7-9. The resulting mechanism would be unnecessarily complicated, and would be operable only in a domain in which none of the variables changes sign. It is much more satisfactory to use as grid generator a star linkage (Chap. 9). This can be so designed as to have an almost ideal grid structure, and, in combination with simple trans- formers, makes up a multiplier useful through a domain that includes both positive and negative values of x% and x,-. It is thus evident that the problem of designing new grid generators with ideal grid structure is one of considerable practical importance; it will be the subject of Chap. 9. 8-6. Choice of a Nonideal Grid Generator. — The number of types of simple and mechanically satisfactory grid generators is rather limited, but the grid structures of these devices can be varied widely by changing design constants. It appears to be practicable to set up an atlas of grid Sec. 8-6] CHOICE OF A NON IDEAL GRID GENERATOR 239 structures from which, by simple comparison with the grid structure of a given function, it would be possible to select a type of grid generator suitable for a mechanization of that function, and to determine approxi- mately the required design constants. We may note here some char- acteristics of this problem, and some methods of simplifying it. The grid structure of a given function may differ from the catalogued grid structure of a satisfactory grid generator for any or all of four reasons : 1. They may differ by a topological transformation, Eqs. (8), which is to be carried out by the transformer linkages. 2. The contours C of the grid structures may not correspond. 3. The centers S chosen for the grid structures may not correspond. 4. The catalogued grid structure may correspond to use of the wrong terminal as output terminal. These four factors will be considered in turn. Regularized Grid Structures. — In order to make possible direct com- parison of the grid structures of given functions and given grid generators, it is desirable to reduce to a common form all grid structures which differ only by a topological transformation. This common form will be termed the " regularized grid structure." It is possible to mechanize a given function by a given grid generator if, and only if, their regularized grid structures are identical, or can be made so by proper choice of the ele- ments mentioned in Items 2, 3, and 4 of the preceding paragraph. In general terms, one may define a regularized grid structure as that obtained from any given grid structure by applying a topological trans- formation which converts the rectangular grid of the original structure into a square grid. More precisely, the transformation to be applied is that which maps into a square grid the very fine grid structure formed in the limit as the contour C approaches the center S. Let the given functional relation be xk = ffa, x3). (14) The transformation to the plane of the new variables {zi, z2) can be defined in terms of line integrals in the (xi, x,)-plane, extending from the chosen center S = (x;o, Xj0, Xko) along the contour of constant as*: Zi - <f>i(xi) = / ( &) dxi, (41a) J XiO \OXi/ Xk=XkO «i = *,(«*) = / *' (f£) dx,-. (416) J XiO \°^}/ Xk=XkO The variable zk is then so defined, as a function of xk, that Eq. (14) reduces to zh - Zi + z-i (42) 240 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-6 along the line Zi = z,- in the (zi} z;)-plane. Rewriting Eqs. (41) as Xi = <t>-l(Zi), (43a) Xi = fr'fe), (436) one may express the required relation as xk ->M?Mf)} (44) All functions with ideal grid structure have the same regularized grid structure — that illustrated in Fig. 8-9 — except for possible differences in the spacing of the grid lines. For instance, if the given relation is xk = (z? + af)M = /(*,, xt) (45) (cf. Fig. 8-10) one has (K\ = *i, (*i\ = IL. (46) Then Thus Xi = 4>~Kz%) = (xk + ZxkoZi)^ (48a) X,- = 0-ife) = (z?0 + 22*,*/)*, (486) and Eq. (44) becomes Xk = (xlo + 2zfcoZfc)^, (49a) or ^ZjfcO As with all functions having ideal grid structure, the z's thus denned satisfy Eq. (42) not only when Zi = z,-, but throughout the domain in which the transformation Eq. (41) is defined and single- valued ; the regularized grid structure is that of Eq. (42). Figure 8-12 shows a regularized nonideal grid structure, that of Eq. (23). It is, in fact, the regularized form of the grid structure shown in Fig. 8-7, and has been constructed graphically by reference to that figure, rather than by analytical working out of the transformation discussed above. We know that the rectangular grid of Fig. 8-7 would be reduced to an almost square grid by this transformation. In Fig. 8-12 this grid has been constructed as exactly square, with negligible error. The con- Sec. 8-6] CHOICE OF A N ON IDEAL GRID GENERATOR 241 tour lines in Fig. 8-12 must have the same relation to this square grid as the contour lines of Fig. 8-7 have to the rectangular grid; points of intersection are easily established by interpolation, and the contours passed through them. Such a construction is quite accurate enough for the purposes here contemplated if the mesh of the original grid struc- ture is not too open. The curves of the structure thus established must correspond to equally spaced values of Z;, zy, zfc that satisfy Eq. (42) along the line Zi = Zj. One can determine the spacing constant a only by reference to the transformation equations; this, however, is a matter of scale which is of no practical importance. In a regularized grid structure the square grid contributes noth- ing to the characterization of the function ; attention can be focused on the system of contours of con- stant Zfc. The usefulness of the idea of regularized grid structure is largely due to this fact. Effect of Change of Contour C. — The transformation to a regularized grid structure converts the functional relation *j *4a \ \ \\ / \ \^j ^> \ k \j ^ X +2a \ \^S \ \£ \ \j ^ X \\ \ \ \ \ \.' k> \ \ Zi -2a ^ s# \ \ s" Xv ' ks \ \ -4.r» ' ^ \ Q >* \ * \ V \\ \ -I [a la 0 +2 a +4a Fig. 8-12. ■Regularized nonideal grid struc- ture. into another, xk = f(xi} Xj) *k = g(zi, Zj), (14) (50) which defines a " regularized surface'' in (zi} Zj, z^-space. This surface is tangent to the plane zk = zt + zy all along the line zk = 0, and intersects it all along the line Zi = z;- ; its form is made obvious by the contour lines of the regularized grid structure. Change in the choice of the contour C changes the spacing of these contours, but not the form of the regularized surface that they describe. Thus, in comparing regularized grid struc- tures of given function with those of given grid generators, one should compare surface to surface, not contour to contour. This can be done without knowledge of the spacing constant. Effect of Change of the Center S. — Passage to the regularized grid structure always transforms the contour B through the chosen center S into the straight line Zi -j- Zj = 0 in the (z», z,)-plane. When the grid structure is ideal, all contours of constant xk are transformed into parallel straight lines; the appearance of the regularized grid structure does not 242 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-6 depend on S. When the grid structure is nonideal this is not precisely true. Adjacent contours are converted into lines which are only approxi- mately straight, and the special characteristics of the nonideal structure become evident in the form of the more remote contours. Choice of the center S' on another contour B' (Fig. 8-12) would make that contour transform into a straight line, and introduce a corresponding curvature into the transformed contour B. If B' is very nearly a straight line in the original regularized grid structure, the change in form of B, and of the rest of the grid structure, will be small. It is thus evident that the chosen center of the grid structure in Fig. 8-12 could be changed within wide limits with little effect on the appearance of the regularized grid structure. On the other hand, a striking change would occur if S" were chosen as the center. It is usually sufficient, for practical purposes, to represent a given grid generator by a single regularized grid structure. The domain of usefulness of the grid generator will be limited by mechanical considera- tions, and it will be natural to choose S near the center of this domain. The domain of a given function to be mechanized will also be specified, and a center S' will be chosen near the center of this domain. If the grid generator is to be useful in mechanizing this particular function the centers S and S' must correspond at least roughly, and the difference between them will not cause large differences in the appearance of the regularized grid structures. Effect of Choice of Output Terminal. — Given a mechanism suitable for use as a grid generator, one might choose any of the three terminals as the output terminal, and might associate the input parameters with the other terminals in two different ways. To each of the six possible ways of using this mechanism as a grid generator there corresponds a different grid structure. The appearance of the grid structure depends principally on which of the terminals is associated with the output parameter Yk) interchange of the input terminals, in their association with Yi and F,-, merely produces a reflection of the grid structure in the diagonal line Yt = Yi. In an atlas of grid structures one might then represent each mecha- nism by three regularized grid structures corresponding to the three choices of output terminal. Alternatively, one might present a single regularized grid structure. For each given functional relation it would then be necessary to construct three regularized grid structures, with Xi, x3; and xk, in turn, treated as the output variable. A match between one of these three structures and a catalogued structure (after a possible reflection in the diagonal) would then show that the catalogued mecha- nism could be used, and would indicate the way in which the parameter should be associated with its terminals. Sec. 8-7] USE OF GRID STRUCTURES IN LINKAGE DESIGN 243 8-7. Use of Grid Structures in Linkage Design. — The concept of grid structure is of fundamental importance as an aid in designing grid generators for special applications. For this purpose it becomes neces- sary to introduce the following generalization of the idea. Generalized Grid Structures. — In the preceding discussion the grid structure of a functional relation xk = f(x{, Xj) - (14) has been defined as a system of lines in the (a:*, a:,-) -plane: straight lines representing constant values of Xi and x3; which form a rectangular grid, and a superimposed family of contours of constant Xk. Such a grid structure is a very special form of intersection nomogram representing the given relation. Now it is not at all necessary to treat Xi and Xj as cartesian coordinates. Instead, one can take the plane of representation as the (z», zj) -plane, and let Xi and Xj be any pair of curvilinear coordinates in this plane, given in terms of the cartesian coordinates Zi, Zj, by Xi = Xi(zi} zi), (51) Xj = Xj[Zi, Zj). The construction of the grid structure then proceeds as before. Adjacent contours B and C of constant Xk are chosen, and between them, beginning at the center S, there is constructed a step structure consisting of portions of contours of constant Xi and Xj. These latter contours, extended through the plane, make up a curvilinear grid, instead of the rectangular one previously obtained. The complete grid structure consists of this grid, together with contours of constant Xk which pass through the grid intersections r = s, or interpolate smoothly between these contours. The same grid structure can be obtained in a different way. Let Eqs. (51) define a topological transformation between the (x^ z/)-plane and the (z*, z3) -plane. This transformation will carry a center Sx in the first plane into a center Sz in the second, and a contour Cx in the first into a contour Cz in the second. It will also transform the entire grid structure defined in the (#;, Xj) -plane, using Sx and Cx, into the grid struc- ture defined in the (zi} z;) -plane, using Sz and Cz. Such a topological transformation of the grid structure will, of course, affect none of its intersection properties ; in particular, it will still serve as an intersection nomogram representing the given function. We shall, in fact, consider grid structures which differ only by a topological trans- formation of the form of Eq. (51) as equivalent representations of a func- tional relation. Mechanical Realization of a Given Grid Structure. — The author's technique for mechanizing functions of two independent variables makes 244 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-7 important use of such topological transformations of grid structures. The basic idea is to transform the given grid structure into a form which suggests a satisfactory mechanical form for a grid generator. The tech- nique employed in this transformation will be indicated in later chapters ; here we shall merely take note of the way in which a given grid structure may suggest a corresponding mechanization of the function. Fig. 8-13. — Grid structure of Xk = Xi/xj (a = 1.25). Consider, for example, the grid structure of the relation xk = Xi Xi (52) as shown in Fig. 8-13. The spacings of the rectangular grid lines change in geometrical sequence; the fixed ratio is here 1.25. The contours of constant Xk are radial lines; the corresponding value of Xk for each line is the value of Xi at its intersection with the horizontal line Xj ■ = 1. At each point of this figure one can read off corresponding values of xi} Xj, and xk which satisfy Eq. (52) . Now, let there move over this figure a pin con- nected mechanically to three different scales. If these connections and scales are so arranged that one can read on the first scale the value of Xi at the position of the pin, on the second scale the value of xJ} and on the third scale the value of Xk, then the device as a whole becomes a mechani- zation of the given function. In the present case, the first scale should show the horizontal displacement of the pin from the origin, the second scale its vertical displacement; the reading of the third scale should be Sec. 8-71 USE OF GRID STRUCTURES IN LINKAGE DESIGN 245 Fig. 8-14. — Transformed grid structure of Xk — x%/xj. 246 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 87 proportional to the horizontal displacement of the intersection of the radial line through the pin with a horizontal line. The divider (or multi- plier) of Fig 1-10 accomplishes this in a very simple and obvious way; it is the natural mechanization of the grid structure of Fig. 1-12, which differs from Fig. 8 13 only by a reflection A topological transformation of this grid structure will carry it into a form (Fig. 8-14) suggesting a very different type of mechanization. The horizontal lines of Fig. 8-13 are transformed into a family of circles, all of the same radius, Li, with centers lying on a straight line C\. The vertical lines of Fig. 8-13 are transformed into a second family of circles, all of the same radius, L2, with centers lying on the curved line Ca. Finally, the radial lines of Fig. 8-13 are transformed into a third family of curves. These are very nearly, although not exactly, circles with the same radius, L3; since the approximating circles intersect at a common point their centers must lie on another circle with radius L3 — curve C3 in the figure. On ignoring the small deviation from circular form of the curves of the third family, we are led directly to the mechanization shown in Fig. 8-15 — an approximate divider or multiplier, but a quite accurate one. The joint P can be made to lie on a particular circle of the first family by placing it at one end of a bar PA i with length Li, and fixing the joint A i in the center of this circle, on line CV Conversely, if the joint Ax is constrained to lie on the line d — as by being pivoted to a slide — it will necessarily be always at the center of the £,-circle on which P lies ; a scale placed along C\ can thus be calibrated to give the value of Xj at the position of the pin P. In the same way, the value of Xi can be read on a scale lying along the curved line C2, using as index point the joint A2 connected to the pin P by a bar of length L2. Finally, values of the quotient Xk might be read on the circular scale C3. Instead of pivoting the bar PA3, of length L3, to a circular slide, one can constrain the point A3 to lie on the curve C3 by a second bar OA3, also of length Rh pivoted at the center of this circle. As shown in the figure, the index point has been transferred to this second bar in an obvious way. The shortcomings of this particular device are obvious: the use of a curved slide, and the nonlinearity of the x{- and zfc-scales. To improve it one should devise a more satisfactory way to guide the point A 2 along the curve C2, and should linearize the Xf and a^-scale readings by transformer linkages, such as harmonic transformers or three-bar linkages. How this can be accomplished is illustrated in Fig. 8-16, which shows the first linkage multiplier so designed as to be operable through a domain including positive and negative values of all variables. The point Ai is constrained to follow the curve C2 of Fig. 8- 15 by placing it on an exten- sion of the central bar of a three-bar linkage afiyb. (The required design technique is indicated in Sec. 10-4.) Motion of Ai along C2 produces a Sec. 8-7] USE OF GRID STRUCTURES IN LINKAGE DESIGN 247 Fig. 8-15. — Mechanization of the grid structure of Fig. 8-14. 248 LINKAGES WITH TWO DEGREES OF FREEDOM [Sec. 8-7] Fig. 8-16. — Linear mechanization of x» = x/x*. Sec. 8-7] USE OF GRID STRUCTURES IN LINKAGE DESIGN 249 corresponding rotation of the bar a/3 of the three-bar linkage; a nonideal harmonic transformer converts this rotation into linearized readings on the ^t-scale. To linearize the x^-scale the rotation of the bar OA 3 has likewise been converted into linear motion of a slide by means of a non- ideal harmonic transformer. This design of a practical linkage multiplier has thus been arrived at in three steps: 1. Topological transformation of a multiplier grid structure into a con- venient form. 2. Design of a simple device for mechanizing this grid structure. 3. Conversion of the design to a more satisfactory form, by applying constraints in a different way and linearizing the scale readings. An important fourth step is the final adjustment of linkage dimensions. These steps are by no means unique, and one can design a great variety of linkage multipliers. For instance, it is possible to find other transforma- tions of the multiplier grid structure in which the curve Ci becomes a circle or a straight line, and the design of a linkage constraint for the point A2 becomes trivial. To accomplish this one needs a thorough understanding of the techniques to be discussed in the next chapter. CHAPTER 9 BAR-LINKAGE MULTIPLIERS A technique for designing bar-linkage multipliers will be developed in this chapter, both for its intrinsic interest and as an example of a general technique. The problem of mechanizing any other functional relation with ideal grid structure is essentially the same, as regards the design of the grid generator; differences arise only in the details of transformer linkage design, which will need no discussion here. Yx x Y2 Fig. 9-1.— Star grid generator. 9*1. The Star Grid Generator. — The grid generator considered throughout this and the following chapter will be the "star grid genera- tor" or "star linkage" illustrated in Fig. 9-1. The general principles to be explained can be applied to other grid generators, but the detailed constructions will of course require modification. The reader will recognize that in the case of the star grid generator these constructions are particularly simple; this simplicity and the satisfactory mechanical properties of this device give it a special usefulness in practice. The star linkage consists of three links, Lh L2, Ls, joined together at one end by a common joint P. The lengths of these links we shall also denote by Lh L2, Lz. At their far ends are free joints Ah Ai, A3, which are in some manner guided along three curves d, C2, Cz. Input and out- put parameters, Yh Y2, F3 can be read at these joints on arbitrarily 250 Sec. 9-2] DESIGNING A STAR GRID GENERATOR 251 graduated scales lying along these curves. The linkage establishes between these parameters a relation F3 = G{Yh F2) (1) which characterizes its behavior as a grid generator. It is at once clear that any functional relation that can be generated by a star linkage can also be represented by an intersection nomogram con- sisting of three families of circles, of radii Lh L2, L3, respectively, repre- senting constant values of the parameters Fi, F2, F3. In fact, the linkage could be used in drawing this nomogram. If the joint A\ is fixed at the point Fi on the Ci-scale, the joint P can then be made to describe the Fi-circle on the nomogram; circles corresponding to definite values of F2 and F3 can similarly be traced out by fixing the joints A2 and Az, respectively. Each circle on the nomogram thus represents a correspond- ing point on one of the three scales. To each configuration of the linkage there corresponds a point on the nomogram at which three circles inter- sect; corresponding scale readings and nomogram points indicate the same triplet of values of the parameters, Yh F2, F3, satisfying Eq. (1). It follows that any functional relation that can be generated by a star linkage must have a grid structure that consists of three families of circles with fixed radii, or can be brought into such a form by the general topo- logical transformation, Eq. (8-51). The grid structure of a star grid generator may be almost ideal over a wide range of parameter values, or strongly nonideal, depending on the link lengths and the choice of curves Ci, C2, C3; it is thus useful in mecha- nizing functions with either ideal or nonideal grid structure. In the present chapter we shall be interested only in designing star grid gener- ators with nearly ideal grid structure. 9*2. A Method for the Design of Star Grid Generators with Almost Ideal Grid Structure. — It will be instructive to examine the grid struc- ture of the star linkage shown in Fig. 9-1. This can be done graphi- cally as illustrated in Fig. 9-2. We choose the center S of the grid structure, and mark the corresponding positions of the joints Ah A2, At, on the three scales with the values of r, s, t, for this center, 0, 0, 0. About these points we draw circles C^\ C^, C{f, with radii Lh L2, L3; these inter- sect at S. Let us choose C(J} as the contour B in the grid structure, thus assigning to F3 the role of xk in Sec. 8-3. Near the point t = 0 on the F3-scale we select another point to correspond to t = 1. About this we describe a circle C(3}, of radius L3, to serve as the contour C of the grid structure. Between the contours C(30) and C(J) we can now construct a step structure consisting of arcs of radius Lh centering on curve Ci, and arcs of radius L2, centering on curve C2. Beginning at S we can follow the circle C(5} to its intersection with the curve C(V. This intersection 252 BAR-LINKAGE MULTIPLIERS [Sec. 9-2 corresponds to the indices r = 0, t = 1 ; in order to satisfy the relation r + 5 = t (2) it must also be assigned the index s = 1. (All three indices are indicated in Fig. 9-2, though any two of them would be sufficient for identification.) At a distance L2 from this point there must lie the point s = 1 on the curve Ct. About the point s = 1 we describe the circle Ca%, which intersects the contour B = C(($ at a point with the indices s = 1, t = 0; the other C,(,) C3to' -Grid structure of a given star linkage. The arms of the linkage are shown with the common joint at the center of the grid structure. index must be r = — 1. At a distance L\ from this intersection there must then lie the point r = — 1 on curve C\. By continuing this process we can build up a step structure between the chosen contours, and estab- lish on the curves C\ and Ci the sequence of points corresponding to integral values of the indices r and s. The families of circles C(i} and C($ about these points form the basic grid of the grid structure. All this follows uniquely from our choice of the center S = (0, 0, 0) and the point t = 1 on Ct. Sec. 9-2] DESIGNING A STAR GRID GENERATOR 253 If the grid structure of this grid generator were ideal, all intersections of the grid with r + s = t would lie on a circle Cf with radius L3 and center on the curve C3. Actually, as shown in Fig. 9-2, it is possible to construct circles C3° that pass very nearly, though not exactly, through these intersections. (Note, for instance, the divergences in the upper right-hand corner of the grid structure.) For practical purposes the grid structure may be considered as ideal over the greater part of the domain illustrated. Within this domain it will mechanize the relation xz = xi + x2, (3) if the scale calibration is that established by this construction, or any other relation with ideal grid structure, if the scales are properly transformed. Figure 9-2 shows very clearly the system of curvilinear triangles which is the distinguishing mark of ideal grid structure. So nearly ideal a grid structure is by no means characteristic of star linkages. This particular linkage has been expressly designed to have an almost ideal grid structure, by a method which will now be described in detail. Our problem is essentially that of constructing three families of circles, Cft Cft C3° (with radii Lh L2, L3, respectively) which intersect to form a triangular structure such as that shown in Fig. 9-2. We begin by choosing arbitrarily six points in a plane. Three of these, Aft Aft A({~1\ will serve as the points r = 1, 0, — 1, on the curve d of the completed link- age; they should lie on a line of moderate curvature, with roughly equal spacings, but can otherwise be chosen at will (cf. Fig. 9-3). The other three points, Aft Aft A(3_1), are to serve as the points t = 1, 0, — 1, on the curve C3, and should be chosen subject to similar conditions. About the points Aft Aft A^, construct circles Cft Cft Cf-» with arbitrary radius L\\ similarly construct about the points Aft Aft A(3-1), the circles Cft £ft C3-1\ with radius L3. Since these circles are to form the basis of the grid structure, L3 should be so large that each C3-circle intersects each circle C\r) in two well-separated points. The intersections of the circles will then fall into groups of nine, well separated in the plane. One of these groups will lie near the center of the grid structure, whereas the other will lie outside the region in which it is almost ideal; it is for this reason that the two groups of intersections should not be close to each other. Choosing one of the two sets of intersections, we label each intersec- tion with the corresponding indices r, s, t: ( — 1, 0, —1), ( — 1, 1, 0), (-1, 2, 1), (0, -1, -1), (0, 0, 0), (0, 1, 1), (1, -2, -1), (1, -1, 0), (1, 0, 1). [The second index is in each case chosen to satisfy Eq. (2).] These nine grid intersections have been chosen with a high degree of arbitrariness; our problem is now to build the grid structure about this nucleus, maintaining its ideal character so far as possible by appropriate choice of the available design constants. 254 BAR-LINKAGE MULTIPLIERS [Sec. 9-2 The three intersections ( — 1,0, -1), (0, 0, 0), (1, 0, 1) must all lie on the circle C(20). By constructing this circle we can establish its radius, L2, and its center, the point A^ on the curve C2. The known points, (—1, 1, 0) and (0, 1, 1), and the known radius L2 then serve to determine Fig. 9-3. — Construction of a star linkage with nearly ideal grid structure. the circle Cp, with center A^; similarly the points (0, —1, —1) and (1, —1,0) determine the circle C2_1) with center A (-D These new circles fix four additional grid intersections: ( — 2, 1, —1), (2, —1, 1), (1, 1, 2), and (-1, -1, -2). There are now determined the three link lengths, Lh L2, L3, and three points on each of the curves d, Ci, C$. By passing smooth curves through these points we can set up a star linkage with a nearly ideal grid structure in the neighborhood of the center S = (0, 0, 0) . To improve the accuracy of this construction, and to extend the domain of nearly ideal grid structure, it is necessary to determine other points on these curves. We now know that the point A (22) must lie on a circle Q22) of radius L2 with its center at ( — 1, 2, 1), and that the point i42~2) lies on circle Qf2~2), with Sec. 9-2] DESIGNING A STAR GRID GENERATOR 255 the same radius and center at (1, — 2, —1). Similarly A{? and A[~2) lie on circles of radius L\ about centers (2, —1, 1) and ( — 2, 1, —1), respectively, and A(32) and A3~2) lie on circles of radius L3 about centers (1, 1, 2) and ( — 1, — 1, —2). No further information is to be extracted from the known points of the grid. Now let us make a tentative choice of the point ( — 2, 2, 0), on the known circle C(30). Together with the known point ( — 1, 2, 1) this deter- mines the circle C(22) and its center A(22); C(22x, in turn, completes the deter- mination of the grid intersection (0, 2, 2). By extension of this process, a tentative choice of the point ( — 2, 2, 0) leads to equally tentative deter- minations of other elements of the grid, according to the following scheme : (-2, 2,0) + (-l, 2, 1)-Aft (0, 2, 2) + (1, 1, 2) - Aft (2,0, 2) + (2, -1,1) -Aft (2, -2,0) + (l, -2, -1)— Af'\ (0, -2, -2) + (-l, -1, -2) -A (-2) 3 ; (-2) C'i> - -* (0, 2; ,2), Cf - -* (2, 0: ,2), Cft - -(2, - -2, 0), cr2> -(0, -2, - •2), C£-» ->(- 2,0, - •2), c(ra ->(-: 2, 2, o; 1. (-2,0, -2) + (-2,1, -l)->ii Thus we arrive finally at a construction for the point ( — 2, 2, 0), with which the whole process was started. This construction will, in general, lead back to the tentatively chosen initial point only if that choice was made correctly. For example, an incorrect choice of the point ( — 2, 2, 0) leads to construction of the dashed grid lines of Fig. 9*3. The curvilinear hexagon, traced out in the clockwise direction, fails to close. A second choice of the initial point leads to a different error in closing ; interpolation finally leads to a correct choice and the construction shown in bold fines. There are thus determined two additional points on each of the curves Ci, C2, Cz, and six additional circles in the grid structure. These six new circles fix additional grid intersections — enough of them, in fact, to determine immediately two more points on each of C\> C2, Cz'. Ap, A[~3), A(i\ A<fz\ A3Z\ A3~s\ For example, the intersection of C(2-2) and C(31} determines (3, -2, 1), and that of Cfx) and C(32) determines (3, —.1, 2) ; these points, in the lower right-hand corner of Fig. 9-3, in turn determine Ai3) and C[Z). It is at this point that the flexibility in the design becomes insufficient to permit construction of an exactly ideal grid struc- ture : these new circles should pass through certain triple intersections, but the construction does not assure that they will do so. For instance, the circles C$+3), C2_3), and C30) should pass through a common point, (3, — 3, 0) ; in Fig. 9-3 it can be seen that they pass very nearly but not exactly through the same point. A similar failure occurs at (3, 0, 3); perfect triple intersections at these points can be obtained only by changing the original arbitrary assumptions. In the present case this would hardly 256 BAR-LINKAGE MULTIPLIERS [Sec. 9-3 be worth while, as the linkage already determined has effectively ideal grid structure in a very large domain. 9*3. Grid Generators for Multiplication. — The example of the preced- ing section should make it clear that it is a simple and straightforward task to design a star grid generator with almost ideal grid structure over a large domain. From a practical point of view this is only a beginning in the work of designing a satisfactory star grid generator for multiplication. Other aspects of this problem will now be indicated. Like any other grid generator with ideal grid structure in an extended domain, the star linkage of the preceding section can be used in designing a multiplier. Calibration of the scales in terms of the variables 2i = At, z2 = ks, z3 = kt (5) will convert it into an adder with very simple structure, mechanizing the relation Zl + 22 = z3 (6) throughout a domain within which each variable may be either positive or negative. Recalibration in terms of variables xh x2, xs, determined by logio Xi = Zi, logio x2 = z2, logio xz = z3, (7) will convert it into a multiplier mechanizing x&2 = xz (8) throughout a domain in which all variables are positive. The most obvious disadvantage of such a multiplier is its use of curved slides. A more generally useful device could be designed if the curves Ci, C2, C3 were of simple, mechanically desirable forms; elaboration of the star linkage into a multiplier like that of Fig. 8-15 or Fig. 8-16 would then follow the lines indicated in Sec. 8-7. One can in fact bring the curves Ci, C2, Cz into desirable forms by making appropriate changes in those elements of the design that were arbitrarily chosen in Sec. 9-2. A satisfactory method for doing this will be indicated in Sec. 9-5, where it can be illustrated in connection with a problem having additional features of interest. Multipliers designed in this way do not permit change in sign of any factor. If xi, for instance, is to pass through 0, then Z\ and r must pass through the corresponding value — 00 . To accomplish this in a mecha- nism with finite travel one must have a Zi-scale of finite length; the sequence of points A{p must approach a point of condensation as r — > — 00 . Even when one has assured the existence of such a point of condensation, corresponding to X\ = 0, it will be necessary to face the problem of extending the scale into the region of negative x%. Sec. 9-3] GRID GENERATORS FOR MULTIPLICATION 257 Such points of condensation have their equivalents in the grid struc- ture of the relation * - x* X\ = — > x2 (9) an alternative form of Eq. (8). In Fig. 94 this grid structure is developed about the center x2 = x3 = I, with the line x\ = 1.25 chosen as the con- tour C. The step structure between the contours B and C approaches the origin in an infinite number of steps; the successively added lines of the rectangular grid will tend to fill out the domain x2 > 0, x$ > 0, but will never extend outside this region. The grid is of course ideal, and includes the contours X\ — (1.25)r, with r taking on all integral values; as r — » <x> these contours approach the hori- zontal axis, and as r — ■> — <x> they approach the vertical axis. To obtain grid structures for this relation in all four quadrants, one must use a separate center for the grid structure in each quadrant. If these four points are chosen in similar positions in the four quadrants, symmetrical with respect to the two axes, and if corresponding contours C are used, then the four grid structures will approach the coordinate axes symmetrically. They will then appear to flow smoothly into each other in crossing these axes, and the whole figure will take on the appearance of a single grid structure (Fig. 9-5). It is important, however, to remember that this result is obtained artificially, and that the coordinate axes are lines of condensation in the grid structure. A topological transformation of Fig. 9-5 that would carry the three families of straight lines into three families of circles of constant radius would convert it into the ideal grid structure of a star linkage. To each circle of this grid structure there would correspond an integral value of r, s, or t, and a calibration point on one of the scales; to the circles obtained by transformation of the coordinate axes there would correspond points of condensation of the scale calibrations. Such a star linkage would thus have the characteristics to be demanded in a grid generator for a multiplier that must allow change of sign in the variables. We shall now use this idea as a guide in designing a very satisfactory star grid generator for multiplication. Fig. 9-4. — Grid structure of x\ = xz/xi. 258 BAR-LINKAGE MULTIPLIERS [Sec. 9-4 9-4. A Topological Transformation of the Grid Structure of a Multi- plier.— Let us attempt to transform the grid structure of Fig. 9-5 (car- tesian coordinates x2, z3Hnto an ideal grid structure in which each of the three families of straight lines in the original structure is represented by a family of congruent circles (cartesian coordinates y2, 2/3). First, let us consider the family of lines of constant x\. In the original grid structure all these lines intersect at a common point 0. Such a property will not be changed by a topological transformation; in the Fig. 9-5. — Grid structure of a multiplier or divider permitting changes in sign of the variables. transformed grid structure the corresponding family of circles of radius L\ must all intersect at a common point 0'. (See Fig. 9-6.) It follows that the centers of these circles must all lie on a circular arc ab with radius L\ and center at 0' '. The radius L\ can be chosen at will, as the problem is independent of the scale of construction; it is usually convenient to take Li as the unit of length. The relation between a given straight line of the original grid struc- ture and the circle into which it is transformed may be established by examining the topological transformation in the neighborhood of the Sec. 94] TRANSFORMATION OF A GRID STRUCTURE U 259 Fig. 9-6. — First topological transformation of the grid structure of a multiplier. 260 BAR-LINKAGE MULTIPLIERS [Sec. 94 origin 0. In its general form this transformation is x2 = x2(y2, yz)A ,1Qv xz = £3(2/2, 2/3).} In the neighborhood of 0, where all variables can be treated as small quantities, this reduces to Xi = C22?/2 + C23?/3, J /-. ]N X3 = C322/2 + C33?/3, ) on neglect of small quantities of the second order (This, of course, is valid only if 0 is not a singular point of the transformation.) Let us assume that the transformed grid structure is symmetrical with respect to the horizontal axis, in the neighborhood of the origin. Then we must have C23 = C32 = 0, (12) and l3 = C3_3^ (13) X2 C22?/2 in the immediate neighborhood of 0. In other words, a line of slope x\ at the origin 0 of the original grid structure is transformed into a curve through the origin 0' with slope changed by the constant factor (C33/C22). Since the slopes of the successive zi-contours in Fig. 9-5 change in a geometric progression, the slopes at 0' of the corresponding circles must also change in geometric progression, and by the same ratio (1.25). This is true also of the slopes of the radii from the origin 0' to the centers of these circles, which are the negative reciprocals of the slopes of the circles themselves. Choosing arbitrarily a value of C33/C22, we can then con- struct, in the transformed grid structure, circles corresponding to each of the lines of constant xi in the original grid structure. In Fig. 9-6 there are indicated four of these circles (Ao, A-i, A-2, A_3) with centers above the horizontal axis at points aQ, o_i, a_2, a_3; the distinguishing subscripts are the r values of the original grid lines, which lie in the second and fourth quadrants of Fig. 9-5. The corresponding lines in the first and third quadrants transform into the circles Bo, BL_i, B-2, i?_3, with centers bo, 6_i, 6_2, b-3, which are the mirror images of a0, a_i, a_2, a_3, in the hori- zontal axis. The sequence of points a0, a_i, a_2, . . . , which lies in the domain of positive X\, has a point of condensation a-x on the horizontal axis; an extension of the £i-scale into the domain of negative Xi is provided by the symmetrically placed sequence bo, 6_i, . . . , which has the same point of condensation. This point is of course the center of the circle into which the vertical axis of the original grid structure is transformed. Sec. 9-4] TRANSFORMATION OF A GRID STRUCTURE 261 Consideration of this family of circles will make it clear that there exists no topological transformation of the type under discussion which maps the whole of the (x*, #3)-plane onto the (y2, 2/3)-plane. The original contours of constant Xi intersect only at the origin and at infinity, but the circles into which we are attempting to transform them may intersect anywhere in the (y2, ?/3)-plane, if arbitrarily large values of r or xi are admitted. We can at best hope to establish a topological transformation that carries a portion of the original grid structure (certainly one within which the magnitude of xi is limited) into a grid structure consisting of arcs of circles. This is quite sufficient for our purposes, since r = — <x> f Xl = 0, are not excluded from the domain of the transformation. Now let us consider the family of circles of constant xz. Since the original lines Xz = c were symmetrical with respect to the horizontal axis, it is natural to give the transformed circles similar symmetry ; their centers must lie on the horizontal axis cd. The vertical axis Xz = 0 is already known to be transformed into a circle of radius L\, with center at A^w. It follows that this second family of circles must have the same radius as the first: L3 = L\. Since the lines xz = xf converge on x3 = 0 as I __> _ co } the point a-M must be a point of condensation on the #3-scale, as well as on the £i-scale. The lines of the original grid intersect the £3-axis at xf ; the transformed circles must intersect the 2/3-axis at points determined by «ff = *s(0, yf)\ (14) or, in the immediate neighborhood of 0', by xf = end?. (15) The values of xzl) go to zero in a geometrical progression (ratio 1.25) as t —> — oo . It follows that the sequence of values y'f approaches zero, and the centers of the x$ circles approach a^^, in a similar progression as t — > — oo . As a first attempt to find a transformation of the desired character, let us assume that this geometrical progression is exact, rather than an approximation valid only in the neighborhood of 0' ; that is, we assume that Eq. (15) is valid for all t. Then, after choosing arbitrarily a value of c33, we can construct the circle corresponding to any z3-line of the original grid structure. In Fig. 9-6 there are shown eight of these circles, with centers c0, c_i, c_2, c_3, and do, d-h d_2, d-z. (The subscript is the index t.) The c's lie in the domain of positive x3, and have a con- densation point at Xz = 0; the d's provide, somewhat artificially, an extension of the scale into the domain of negative Xz. The assumptions made up to this point [transformation of the Xi- and ^3-contours into families of circles symmetrical to the horizontal axis, validity of Eq. (15), and special values of c22 and c33] determine two 262 BAR-LINKAGE MULTIPLIERS [Sec. 94 families of intersecting circles, and thereby determine completely the nature of the topological transformation. It remains to be seen whether this transformation has the desired character — whether the z2-contours are also transformed into a family of circles with common radius L2. It is immediately evident that this is not the case. In Fig. 9-6 there appear 64 points of intersection of the Xi- and z3-contours, distinguished by small circles. These are the transformed positions of the intersections in the original grid, and through them must pass the transformed contours of constant z2. It will be observed that the intersections in the lower half of the grid he on curves that are concave upward, whereas those in the upper half he in curves (not shown) that are concave downward. By the symmetry of the construction, the straight line z2 = 0 must be trans- formed, not into a circle, but into the straight line 2/2 = 0; the radii of curvature of the other z2-contours increase as they approach this limiting straight line. The transformed ideal grid structure is not that of a star linkage ; indeed it is evident that if such a grid structure exists it must be an unsymmetrical one. We can, however, approximate this grid struc- ture by the nonideal grid structure of a star linkage, replacing the system of z2-contours of the ideal grid structure by a system of approximating circles of the same radius. This " approximately ideal" grid structure, which does have the other characteristics that we desire, will later be made much more nearly ideal by readjustment of the design constants. It is possible to pass circles very nearly through all the intersections in the lower half plane of Fig. 9-6, by choosing a mean value L2 for the radius of curvature and locating the centers of the circles in the upper half plane. This will establish the general position of z2-scale, and will make it necessary to pass through the intersections of the upper half plane circular arcs that are concave upward; the fit there cannot be very good, and we must split the errors of construction as well as possible. The best way to do this is to construct a circle through one set of intersections in order to establish a radius L2. (In Fig. 9-6, E0 is the circle in question, and the radius chosen is just equal to Li and L3.) With this radius, we construct arcs about each of the known grid inter- sections. If the grid structure under construction were to be ideal, the arcs characterized by a given value of s = t — r would all intersect in a common point. This is not the case here ; instead of points of intersection there exist more or less diffuse regions of intersection, within which we can locate the centers of the grid circles with some degree of arbitrariness. This arbitrariness can be used to good advantage. If a simple mechani- zation of the grid structure is to be possible, the z2-scale must he in a simple curve, preferably a straight line or a circle. In the present case, the regions of intersection lie roughly on a circle. In particular, the circular arc e, with center at Q, passes nicely through all regions of inter- Sec. 9-4] TRANSFORMATION OF A GRID STRUCTURE 263 section except for three at the extreme ends. This circle will be taken as the #2-scale on this scale, and as near to the centers of the regions of intersection as is possible we choose the centers eh e0, e_i, ... of the grid structure circles Eh E0, 2?_i, ... in the lower half plane, and the centers /i, /o, /-i, . . . , of the grid structure circles Fh F0, F_h . . . , in the upper half plane. The grid structure circles must converge on a First approximate multiplier. circle E-w = F-w through the origin 0'. The center of the circle E_M is e-^ hs f_ao} the point of condensation of the point sequence es in the positive domain of x2, and of the point sequence /s in the negative domain, as s — > — oo ; it is the zero point on the x2-scale. This completes the determination of the constants of the star linkage. The grid structure of Fig. 9-6 is mechanized by the approximate multiplier sketched in Fig. 97. This consists of a star grid generator 264 BAR-LINKAGE MULTIPLIERS [Sec. 9-5 with arms Lh L2, Lz, meeting at a common joint P. The free end of Lx is forced to move along a circle a with center 0' , and the free end of L2 along a circle e with center Q, by arms Ri and i?2, respectively; the free end of Lz moves in a straight slide c. It follows from the theory of the transformation that the lengths of Llf Lz, and R i must be equal; L2 also has this length, but only accidentally. To use this device as a multiplier, the scales must be calibrated in terms of the variables Xi, x2, Xz, related to the indices r, s t, by Eqs. (5) and (7). The scale points of Fig. 9-6 thus occur for values of xh x2, x$, which change in geometric progression; these are the scale points shown in Fig. 9-7. One can easily show that xi = Kx tan 0, (16) xz = Kzh. (17) Calibrations on the z2-scale follow (though not uniquely, since the multiplier is not exact) from the relation xz = x&i. The reader should sketch this mechanism for xi = 0 and for x2 = 0, in order to see why in these cases the value of x z is necessarily zero. He will also be able to show by simple geometry why the multiplication is almost exact for small values of x\ and x2. 9*6. Improvement of the Star Grid Generator for Multiplication. — The errors in the multiplier of Fig. 9-7 appear very clearly in its grid structure, in which many of the triple intersections characteristic of ideal grid structure have disintegrated into small triangles. To improve this design we must change one or more of the families of grid circles in such a way as to reduce the size of these triangles. This can be done by a method that is useful whether the function to be mechanized has ideal or nonideal grid structure. Let us examine the possibility of improving the grid structure of Fig. 9-6 by changing the family of circles CD, while keeping fixed the families AB and EF. In Fig. 9-8 there are shown eight of the circles A B and 10 of the circles EF, defining 80 points of intersection through which circles CD should pass. It has already been noted that the circles CD must have the same radius as the circles AB; to improve the grid structure we can change only the positions of their centers. Let us attempt to do this by the method of the preceding section, drawing arcs of radius L3 about the 80 intersections of the grid. As before, the arcs with a common index t do not intersect in a common point, as they must if an ideal grid structure is to be obtained. Instead, we observe an interesting and characteristic phenomenon : arcs with centers in the same quadrant of the grid structure intersect nicely in a series of points that define a curved ^3-scale — but a different scale for each quadrant. This is due to the fact that the four quadrants of the original grid structure are actually independent, and have been associated with each other in a Sec. 9-5] IMPROVEMENT OF A STAR GRID GENERATOR 265 symmetrical, but essentially artificial, manner. In Fig. 9-8 the points corresponding to positive x2 have been marked with small circles, the others with dots. /_; /- <£.■/. Fig. 9-8. — First step in redesigning the multiplier grid structure. It is evident that we can add circles CD to produce a grid structure that is nearly ideal in various pairs of quadrants — those with x<i > 0, or with x-i < 0, or xi > 0, or Xi < 0 — but that it will then be far from ideal in the other pair of quadrants. To state it differently, we can 266 BAR-LINKAGE MULTIPLIERS * [Sec. 9-5 combine the scale segments of Fig. 9-8 in several ways to form an z3-scale, obtaining a multiplier that will be very accurate so long as one of the factors has a particular sign, but much less accurate when it has the other sign. On the other hand, the difference between the diverging scale segments is too great to permit a satisfactory compromise; no choice of form for the z3-scale can make the grid structure nearly ideal in all quadrants. To obtain an ideal grid structure the circles AB and EF must be so chosen that the diverging Xz-scale segments coalesce to form a single xz-scale. This method of stating the problem reduces it to an especially convenient form, which can be solved by alternately adjusting the circles AB and EF. We shall choose to change first the family of circles EF. In varying the grid structure there are two important principles to be observed : 1. The grid structure should not be given more than one degree of freedom at a time. 2. The grid structure should not be excessively sensitive to changes in the varied parameters. This can often be assured by changing parameters in such a way that certain elements of the grid structure remain unchanged. For example, to improve the grid structure of Fig. 9-8 let us rearrange the circles EF without changing their common radius. Let us maintain the circular form of the x2-scale, but rotate it about the point e_x = f_w, thus keeping unchanged the circle E^ = F-^. This rotation gives the one degree of freedom that we desire in the problem, according to Principle 1; we must therefore remove all freedom in the calibration of the scale. By Principle 2, the rule for this calibration must be such that the grid structure changes only slowly with rotation of the x2-scale; we shall therefore demand that it keep unchanged the grid intersections marked with bold dots in Fig. 9-8. Let rotation of the £2-scale, e, about the point €_«, carry it into the position e' (Fig. 9-8). The calibration points e's, fs on this new scale must lie at distance L2 from the corresponding fixed grid intersections, and are easily constructed. The new system of grid circles E'F' can then be drawn, and finally, by constructing arcs about the new grid intersections, the new set of £3-scale segments. Thus the whole construction does have one degree of freedom, and it is easy to study the effect on the form of the z3-scale segments of rotating the z2-scale. Trial will show that rotation of the scale in a clockwise direction brings closer together the two z3-scale segments to the right of a-^; by an interpolation or extrapolation one finds a rotation of e which makes the separation of these scale segments very small. This is the rotation shown in Fig. 9-8, the tentatively chosen Sec. 9-5] IMPROVEMENT OF A STAR GRID GENERATOR 267 rotation having been omitted as of little interest. The new scale and the new circles E'Ff are shown in Fig. 9-9, together with the new form of the a;3-scale construction. The improvement in the agreement of the ^3-scale segments on the right is very striking, but adjustment of the circles AB will be required to improve the agreement on the left. Fig. 9-9. — Second step in redesigning the multiplier grid structure. To improve the grid structure further let us rearrange the circles AB without changing their radii. To do this we shall keep fixed the form of the ^i-scale, ab, while rotating it about the point a_M. Calibrations on the new scale will be held at a fixed distance L\ from the grid inter- sections indicated by bold dots in Fig. 9-9. With these changes we must make one other change, which has no parallel in the step previously described. Rotation of the Xi-scale will move its center from 0' to 0". 268 BAR-LINKAGE MULTIPLIERS [Sec. 9-5 The new circles A B will all intersect at this point, through which there must also pass the convergence limit of the circles EF, ^_oe. It is therefore necessary to~keep L2 always equal to the distance from e-M to 0" as the X\ scale is rotated. We have then to consider a simultaneous variation of both the circles AB and EF, but it is a variation with one degree of freedom which offers no difficulties. Fig. 9-10. — Improved multiplier grid structure. Beginning with a trial rotation of the Zi-scale into a position a'b' (Fig. 9-9), one can establish calibration points a'r, b'r by drawing arcs about the chosen points of the original grid. The AB circles can then be constructed, the new L2 determined, and the EF circles constructed. Finally, the new z3-scale segments can be established, and the best angular position for the £i-scale determined by an interpolation or Sec. 9-5] IMPROVEMENT OF A STAR GRID GENERATOR 269 extrapolation. In Fig. 9-10 this construction is shown for an zi-scale determined by such an interpolation.1 The four rr3-scale segments do not Fig. 9-11. — Multiplier grid structure with circular cc3-scale. merge exactly, but the groups of arcs make acceptably sharp intersections, through which a straight line can be laid by a small sacrifice in the fit at the extreme right end of the scale. The circles CD constructed about 1 It will be observed that, because of the change in L2, all grid intersections have been shifted, even those used in constructing the new xi-scale. (Old positions are shown by bold dots in Fig. 9- 10.) This is not a matter of importance; all that is required of the construction is that it shall not shift the grid structure too violently. 270 BAR-LINKAGE MULTIPLIERS [Sec. 9-5 points on this rectified z3-scale are shown in Fig. 9-10. In view of the small number of steps required, the result can be considered very satis- factory: the grid structure is so nearly ideal over a wide domain that analytical methods can be employed for its further improvement. It is evident that the grid structure of Fig. 9-10 is not the only possible solution of our original problem. The grid structure of Fig. 98 could Fig. 9-12. — Multiplier grid structure ideal through a very large domain. have been varied in many ways other than those described above to obtain even larger domains of nearly ideal grid structure, and different forms of the scales. Indeed, this method is as useful for control of the form and extent of scales as it is for the improvement of grid structures. For instance, the linear z3-scale of Fig. 9-10 is easily converted into the circular arc shown in Fig. 9*11. This form was obtained by rearranging the circles E'F', this time by increasing the radius of the z2-scale, while Sec. 9-6] DESIGN OF TRANSFORMER LINKAGES 271 keeping fixed the points e'-^ and e'_i. New scale calibrations were so chosen as to keep fixed the grid intersections indicated by bold dots in Fig. 9-11. This figure shows also the new family of circles EF, and the resulting circular form of the a:3-scale; the curvature of this can be changed at will by choosing a new center Q" for the £2-scale. A more elaborate series of variations leads to the grid structure shown in Fig. 9-12, with nearly straight scale. The outstanding characteristic of this grid structure is the large domain within which it remains effectively ideal. K*H Fig. 9-13. — Improved multiplier, with the grid structure shown in Fig. 9-10. 9*6. Design of Transformer Linkages. — The grid structure of Fig. 9- 10 suggests the design for a multiplier shown in Fig. 9-13. The structure of this multiplier is the same as that of Fig. 9-7; increased accuracy has been obtained by changes in the dimensions, but the number and relations of the elements remain unchanged. The ranges of the variables can be changed with some freedom: the readings on the Xy, x%-} and a^-scales can be changed by factors a, b, and c, respectively, provided only that ab = c. (18) 272 BAR-LINKAGE MULTIPLIERS [Sec. 9-6 Such a multiplier, with nonuniform scales, is of limited interest; the real importance of this device lies in the possibility of using it to drive a computer. In such an application it may be regarded as an ideal grid generator and used together with transformer linkages in the linear mechanization of the relations xz = X1X2, (19) xz = fiMfifa), (20) or indeed of any functional relation with ideal grid structure. If a linear mechanization of Eq. (19) is required, the function of the transformer linkages may be regarded as that of replacing the nonuniform scales of Fig. 9-13 by uniform scales; in other cases the transformer linkages serve also as function generators. Let us consider first the problem of designing a multiplier with uniform scales. To describe the configuration of the grid generator we may use internal parameters Yh Y2, Yz, defined in Fig. 9-13. The scales shown in the figure establish a nonlinear relation of the variables x to the parameters Y, Xr = (Xr\Yr) • YT, T = 1, 2, 3, (21) which can be determined, for instance, by measurement of the figure. We wish now to establish the same relation between the parameters Y and the variables x as indicated on uniform scales. We introduce transformer linkages which present new terminals, described by external parameters Xh X2, Xz. These external parameters are related to the internal parameters by the linkage equations Xr= {Xr\Yr)-Yr r=l, 2, 3, (22) and to the variables xh x2, Xz, by linear relations, xr = 40) + Kr(Xr - X?>), (23) which may be symbolized by Xr = (xr\\Xr) • Xr. (24) Our problem is to find linkages such that the linkage operators satisfy the relations (Xr\Yr) = (Xr\\Xr) " (Xr|Fr), T « 1, 2, 3. (25) This problem takes on a completely familiar form when it is expressed in terms of homogeneous parameters and variables: 6h 02, 03, corresponding to Yi, F2, F3; Hh H2, Hz, corresponding to Xh X2, Xz] hh h2, hz, cor- responding to Xi, x2, Xz. In terms of homogeneous parameters and varia- bles a linear transformation reduces to the identical transformation, and Eq. (25) reduces to (hr\6r) = (Hr\6r). (26) Sec. 9-6] DESIGN OF TRANSFORMER LINKAGES 273 Our problem is thus to find linkages with operators {Hr\Br) having the known form of (hr\Br), subject to the condition that the input parameters Yr have a given character — that they are, for instance, angular parameters with a specified angular range. This is exactly the type of problem dis- cussed in Chaps. 4 to 6. As an example of the use of the same grid generator in linearly mecha- nizing another functional relation with ideal grid structure, we may con- sider the problem of mechanizing Eq. (20) with linear scales in xh x^, x$. Table 9-1. — Characteristics of the Scales of Fig. 9-13 xi/a Yi, degrees hx 0i -1.000 -22.3 0.000 0.000 -0.800 -17.6 0.100 0.106 -0.640 -14.0 0.180 0.187 -0.512 -11.1 0.244 0.252 0.000 0.0 0.500 0.502 0.512 11.1 0.756 0.752 0.640 13.9 0.820 0.815 0.800 17.4 0 900 0.894 1.000 22.1 1.000 1.000 x2/b Y2, degrees h2 02 -1.000 -33.7 0.000 0.000 -0.800 -27.8 0.100 0.071 -0.640 -22.6 0.180 0.132 -0 512 -19.9 0.244 0.165 0.000 0.0 0.500 0.403 0.512 19.9 0.756 0.642 0.640 25.2 0.820 0.705 0.800 32.1 0.900 0.787 1.000 49.9 1.000 1.000 x3/c Y3 h3 03 -1.000 -0.264 0.000 0.000 -0.800 -0.198 0.100 0.148 -0.640 -0.148 0.180 0.260 -0.512 -0.112 0.244 0.341 0.000 0.000 0.500 0.592 0.512 0.098 0.756 0.811 0.640 0.120 0.820 0.861 0.800 0.147 0.900 0.921 1.000 0.182 1.000 1.000 274 BAR-LINKAGE MULTIPLIERS [Sec. 9-6 The quantities to be multiplied are Zi = fi{Xi), 22 = /2O2). (27) The x{- and £2-scales of Fig. 9-13 are then to be interpreted as scales of Zi and z-i ; the relation of x\ to Yx and x2 to Y2 follows from the observed relations of Z\ to Y\ and z2 to F2, together with Eqs. (27). Except for this difference of detail in establishing the form of the operators (xr\Yr), the procedure of the preceding paragraph applies without change. The com- pleted mechanism may be of exactly the same type as the multiplier 1.0 0.9 0.8 0.7 0.6 K 0.5 0.4 0.3 0.2 0.1 / / / / / / (hz\02)^y / X- ■■e .y£ h3\e3) z \hx\ei> 0.1 0.2 0.3 0.4 0.5 0, 0.6 0.7 0.8 0.9 1.0 Fig. 9-14. — Operators (hr\dr) for the multiplier, Fig. 9-13. itself, or of even simpler form. It is, indeed, one of the important virtues of bar-linkage generators of functions of two independent variables that their complexity does not necessarily increase with the complexity of the analytic form of the function, as its does with conventional computers. This fact will appear most clearly in the next chapter. As an example of the form of the linearizing operators, we may con- sider those needed for the multiplier in Fig. 9-13, if the ranges of motion Sec. 9-6] DESIGN OF TRANSFORMER LINKAGES 275 are to be those indicated in that figure. Table 9-1 shows the value of each variable at the indicated points of calibration, the corresponding value of the associated parameter, andthe homogeneous variables h and 0 at each point. The operators (hr\0r) are plotted in Fig. 9-14. It will be noted that the operator (^2^2) shows an appreciable discontinuity in slope when h2 = 0.5. This is due to the still imperfect match between the four quadrants of the transformed nomogram. The type of linkage used for each transformer will depend both on the nature of the function and on the type of output terminal desired. In Fig. 9-15. — Multiplier permitting change in sign of only one factor. general it will be found that the linkages discussed in Chaps. 4 to 6 offer all the flexibility required. Examples are provided by Fig. 8-16, which has already been explained, and by Fig. 9-15. The latter figure shows a multiplier designed to permit change in sign of x2, but not Xi. The central joint of the star grid generator appears a little below the center of the figure. The bars have their other ends guided along a horizontal straight line, which serves as the #i-scale, and along circles with centers 0, 0', respectively. Rotation of the end of one bar about 0 is transformed into horizontal slide motion proportional to x2 by an ideal harmonic transformer; rotation of the end of the third bar about 0' is transformed into a parallel slide motion proportional to x3 by a combination of a three-bar linkage and an ideal harmonic transformer. The design 276 BAR-LINKAGE MULTIPLIERS [Sec. 9-6 involves four dimensional constants subject to arbitrary choice: A, which determines the scale of the grid generator and the length of the £i-scale; B, which determines the scale of the three-bar linkage; C and D, which determine the lengths of the x2- and z3-scales, respectively. If X\, X2, X3 are displacements measured in the same units, the linkage generates the relation ^ = 3.0909 ^ ^2. (28) The first steps in designing transformer linkages will involve the use of tabular or graphical methods. A graphical presentation of the oper- ators, as in Fig. 9-14, will make it possible to read off values of the operators for evenly spaced values of 0r, when these are required. [As a rule one should disregard irregularities in the linearizing operators, such as are shown by (h2\62) in the example above; the operators should be replaced by smoothly varying approximations.] Such graphical inter- polations are not necessary when the geometric method is to be used in designing a three-bar linkage. One then needs to use a geometric series of values of one of the variables involved — such as are provided in Table 9-1 in the case of the variables xh x2, x3. To apply the geometric method directly to the entries in such a table one would require an overlay constructed for the same geometric ratio (here g = 1.25). The overlay would also need to be extended in both directions from the zero line by addition of a new series of lines with spacing [cf. Eq. (5-92)] yw-i) _ yco = -agt} t = 0, 1, • • • . (29) The details of this extension of the geometric method will present no difficulty to the reader. To minimize the accumulation of errors in designing transformer linkages, the following procedure is often useful. When two of the trans- formers have been designed, a graphical recalibration of the third of the original scales can be carried through before its linearizing transformer is designed. For example, let new uniform scales for X\ and x2 be con- structed and accepted as exact. On these scales lay down geometric series of points X& = ±C2g\ (30) Next, construct corresponding calibration points on the original non- uniform Xi- and rc2-scales, using the known constants of the transformer linkages. Then, using the known constants of the grid generator, con- struct the points z(3±8) on the original z3-scale that correspond to xx = x[± and x2 = x2% for various choices of r and s. If the grid generator were exactly ideal, and the transformer linkages were without structural error, Sec. 9-7] ANALYTIC ADJUSTMENTS OF LINKAGE CONSTANTS 277 all the points with r + s = t (31) would fall together on two points of the x3-scale, corresponding to xf± = ±C1C2gt. (32) Actually there will be some scattering of these points, and it will be necessary to choose mean positions for the new calibration points x$. on the #3-scale. This method of constructing the z3-scale introduces a partial correction for all design errors committed, up to this point; it remains only to design the transformer linkage for the £3-terminal. 9«7. Analytic Adjustment of Linkage Multiplier Constants. — Final adjustment of the constants of a multiplier can be carried out by analytic mathods similar to those described in Chap. 7. From the point of view of theory, the present problem differs from the earlier one principally in the necessity for controlling the structural error in a two-dimensional, rather than a one-dimensional, domain. From a practical point of view, the large number of adjustable constants makes a complete treatment of the problem tedious, but assures high accuracy in the result if enough care is taken. The present section will describe a straightforward application of analytic methods to the final adjustment of linkage con- stants; the next section will indicate some associated or alternative techniques used by the author. The reader should be warned that the practical importance of this part of the design procedure, and the labor required, are out of proportion to the brief discussion that can be devoted to it in this volume. If the combination of star grid generator and transformer linkages were an exact linear multiplier, it would generate a relation RXS = X,X2 (33) between external parameters Xi, X2, X3, at least within a domain Xim ^ Xi ^ Xim, X2m ^ X2 ^ X2M, X3m ^ X3 ^ X3A/. (34) Here R is a constant, and the parameters are so defined that Xi = 0 when xi = 0, etc. Because of structural errors in the mechanism it will actually generate a relation that can be written as RXZ = XiX2 + 8(Xh X2). (35) The last small term is the structural-error function of the mechanism, which must be brought within specified tolerances by adjustment of the linkage constants. In the case of a multiplier it is convenient to gauge the error of the mechanism by the structural-error function. The effort in the calculation can be decreased by computing the error function for spectral values of X({>+ = X[°> •gr, xp- = -x^ 9r, X(2S,+ = x^ 9s, X'?- = -XT 9s, 278 BAR-LINKAGE MULTIPLIERS [Sec. 9-7 X\ and X2 which form geometric progressions in the two halves of each input scale: = • • • , -2, -1, 0, 1, 2, • • • , (36a) = • • • , -2, -1,0, 1, 2, • • • . (366) If the multiplier were exact, the corresponding spectral values of X* would form similar geometric progressions in either half of the output scale : RX[r+s)+ = XTXTgr+s = RX(T9r+s, (37) RXir+s)- = -XTXTgr+s = -RXTgr+s. The spectral values of X3 in either half of the scale would then depend only on the value of r + s for the corresponding spectral values of X\ and X2. With the actual multiplier we have instead flXjp±.«±> = Xp±XP* + KXF* X(2S)±); (38) the spectral values of X3 do fall into groups according to the value of r + s, but they are scattered about the ideal value for the group, Xpgr+a, with errors given by the structural-error function d(X[r)±, X(2S)±). The spectral values of the structural-error function are conveniently arranged as a matrix — or, more properly, as an assembly of four infinite matrices. To simplify the notation we shall write - B(XP ± X2*>±) = «*± (39) With value of Xi increasing upward, values of X2 increasing to the right, the matrix takes the following form: Etr Et.r e±{;, £±fo #±i7-i : Etx '.Etx '•Ef±x • E±,U Ett Etf E±£0 Et.t Et,t «ti!i E-»\\ . E-:°. . . Frrr.1 : *rn-. : E±t-\ E±to E±t:' ' E~o,r E^tQ E^~i ; Eq±„ ; Eq+i EqJ E^f Ej,r Ej* Et,zi ;EtX :ez^ etj ET,t Each row or column of dots represents an infinite number of rows or columns of spectral values of X\ as r — ♦ ± » , in the case of rows, or of X2 as s — > ± oo f in the case of columns. It is, fortunately, not necessary to give detailed consideration to these parts of the matrix. In the graphical process of constructing an ideal grid structure it was sufficient to con- sider grid lines with small positive and negative values of r and s, and Sec. 9-7] ANALYTIC ADJUSTMENT OF LINKAGE CONSTANTS 279 the circles of convergence, r = — <» , s = — °o ; the same restrictions on r and s can be made in the present discussion, and for the same reasons. If the grid structure is ideal at Xi = X2 = 0 — and it should be kept so throughout the work — the output parameter X3 will be independent of Xi when X2 = 0, and independent of X2 when Xi = 0. All entries in the central cross of the matrix (40) will then have the same value, E±± . -t-/— 00,— 00* The elements of the structural-error matrix are functions of all structural constants of the multiplier: the dimensions of the star grid generator and the transformer linkages, the origins from which the parameters Xh X2, and X3 are measured, and the constant R of Eq. (38). Let the independent constants be n in number: gi, g2, . . . gn. A small variation Agt of the constant gi will change the matrix element E?s± by an amount _r's • A^-. It will then modify the matrix E by cl- adding to it Agi times the infinite matrix If small variations are made in all the constants gi, the structural-error matrix will become, to terms of the first order in the A<7», I E + 5E = E + > G,A^. (42) i The problem is then to determine the form of the matrices Gi and to choose small (cf. Sec. 9-3) values for the Ag{ which make the final struc- tural-error matrix E + 5E as small as possible — or at least, to reduce the errors in certain regions of this matrix until they meet specified tolerances. The labor involved in solution of this problem is considerable, and the work must be arranged with care. It is necessary to consider only the portion of the matrix that corresponds to the domain of action of the multiplier. The central cross of the matrix must be included, but the calculations need not be extended to large negative values of r and s> particularly if variations of the constants are restricted to those that maintain the accuracy of the multiplication for Xi = 0 and X2 = 0. Analytic determination of the derivatives (dE±±)/(dgi) is advantageously replaced by large-scale graphical constructions to determine the matrices m Ag{ = GiAgi (43) for small changes Agi in each parameter; these matrices can be used directly in the calculations, or converted into the matrices Gi, as desired. 280 BAR-LINKAGE MULTIPLIERS [Sec. 9-7 The matrices associated with changes in constants of the terminal linkages have simple forms. Variation in a constant gt of the output transformer linkage produces a change in X3 which depends only on X3; consequently the corresponding matrix Gi has identical entries along each line of constant X3. These are lines of constant r + s: diagonals parallel to the principal diagonals in the lower-left and upper-right quadrants (positive X3), and diagonals perpendicular to these in the upper-left and lower-right quadrants (negative X3). All entries in the central cross of the matrix will be identical. A matrix G; associated with a constant gi of the Xi-transformer linkage has entries which in each row are proportional to X2; if ft is a constant of the X2-transformer linkage, Gi has entries which in each column are proportional to Xi. In either case, the entries in the central cross will vanish if the changes in the transformer linkages do not affect the accuracy of the multipli- cation for X\ = 0 and X2 = 0. In varying the dimensional constants of the star grid generator one should follow the principles discussed in Sec. 9-5; one should make changes with one degree of freedom which maintain the invariance of properly chosen elements of the grid structure, and particularly the exact performance of the multiplier for Xi = 0 and X2 = 0. Such changes can of course be described by a single parameter ft, which may be termed a "restricted parameter." It would be extremely difficult to compute by analytic methods the matrices Gi associated with restricted parameters; the graphical construction is quite practicable if the work is done on a sufficiently large scale. The entries in the central cross of these matrices are all zero. It remains to make a linear combination of the matrices E and Gt-, i = 1, 2, • • ■ n, that will have all elements as small as possible within the domain of interest. Following the ideas of Sec. 7-3, one can make n preselected elements of the residual-error matrix vanish exactly. (The solution will of course be spurious if large Aft are required.) One can also apply the method of least squares (Sec. 7-6). The author prefers to build up the required linear combination in a succession of steps, in which there are formed linear combinations of the Gi that can be used to reduce the elements in one or another of several regions of the error matrix E without introducing new errors elsewhere. For simplicity, let us divide the domain of interest into two regions, A and B. By laying out the matrices Gi, one can see how to make a number of linear combinations of these which are small in some part of region A. By linear combinations of the resulting matrices one can build up combi- nations of the Gi which are small throughout region A but large in region B; these can be used to reduce the elements of the error matrix in region B without introducing new errors in region A. Similarly, one Sec. 9-8] GAUGING THE ERROR OF A GRID GENERATOR 281 can build up other linear combinations of the G; which can be used to reduce the errors in region A without introducing new errors in region B. The problem has thus been divided into two simpler and essentially independent problems: that of reducing the error in region A, and that of reducing the -error in region B. If these problems are not simple enough to be solved by inspection, each region can again be subdivided, and the process of forming new linear combinations carried through as before. The method requires some flexibility of approach, and the designer will profit from experience. The author finds it a completely satisfactory method. 9*8. Alternative Method for Gauging the Error of a Grid Generator. — It is usually satisfactory to carry out the final analytic adjustment of dimensional constants for the grid generator and transformer linkages separately. This greatly simplifies the calculations by reducing the number of dimensional constants that must be varied simultaneously. In adjusting the constants of a grid generator it is convenient to use an alternative method for gauging the errors in the almost ideal grid structure. We have noted that, in an ideal grid structure, systems of contours of constant xi, x2, xs meet in exact triple intersections, whereas in a nonideal grid structure these nodal points of the grid disintegrate into little triangles. To make a star grid generator ideal one would have to make these triangles vanish throughout the domain of interest. The linear dimensions of these triangles have the essential characteristics required of a gauge of the error in the grid structure: they change pro- portionally to small changes in the dimensional constants of the grid generator, and vanish when the error vanishes. Since they are especially easy to determine by graphical construction, it is very convenient to use them directly as gauging quantities in the final adjustment of linkage constants. The star grid generator is intended to establish a relation Rxz = xix2 (44) between variables X\, x2, x$, read on associated scales which are in general nonuniform. Without attempting to control the uniformity of the scales, we shall attempt to make this relation as exact as possible by varying the constants of the grid generator and the calibration of the scales. As in Sec. 9*7, we choose spectral values of the variables which form geometric progressions: a#>± = ±x[0)gr, ~(S)± _ _L -(0)^8 x,«± _ ± XlX*. g, 282 BAR-LINKAGE MULTIPLIERS [Sec. 9-8 First we may construct the X\- and z2-contours in the domain of interest. The intersections in the resulting curvilinear grid can be labeled with the index pair {r±, s±). Next we construct the ^-contours specified by Eq. (45) . These will not, in general, pass through the inter- sections of the original grid, but will serve to complete small triangles with one vertex at each intersection. Let the side of the (r±, s±)- triangle opposite the (r ± , s± ) -intersection have length L±± taken as positive if the x3-contour passes the intersection on the side of increasing Xz, negative if it passes on the other side. The deviation of the grid structure from the ideal can then be represented by a matrix L = [Ltf], (46) identical in structure with the matrix of Eq. (40), except that the quan- tities E±± are replaced by the quantities L±f . A change A^ in a restricted parameter gt of the grid generator will change the grid structure; the lengths L±± will become, to terms of the first order, m + M# - 1# + *t? Affc. (47) The quantities 8L±f can be graphically determined for some small Agr»; it is then a simple matter to write down the matrix » - m (48) which corresponds to the matrix G; of Sec. 9-7. If each of the restricted parameters of the grid structure is changed by a small amount, the matrix of triangle dimensions will become, to terms of the first order, L + 5L = L + ^ HiAgr,-. (49) To make the grid structure ideal one would like to choose values of the A<7i (necessarily " small ") that make the matrix L + 5l_ vanish identically. Determination of the Agt can then proceed as described in the preceding section, except that one will not in general attach the same relative importance to reduction of the various quantities L±± as to reduction of the corresponding output errors E±£; what weighting factor is to be applied will be obvious from inspection of the grid structure. When the scales associated with the grid generator have been deter- mined, it will remain to design the transformer linkages and to adjust their constants as described in Chap. 7. Finally, the performance of the complete mechanism must be determined by exact calculation. The procedure described in this section was applied in designing the multiplier Sec. 9-8] GAUGING THE ERROR OF A GRID GENERATOR 283 illustrated in Fig. 9- 15. The residual errors are shown in Table 9-2, which gives Xz — xix2 for a series of values of Xi and x2} when the con- stants A, C, D of Eq. (28) are so chosen that the generated relation should be x3 = x\x2. Table 9-2. — Structural Error xz — X\X2 -in Multiplier, Fig . 915 ^\^^ Xi x2 ""^\ 0.000 0.401 0.511 0.660 0.802 1.000 1.000 0.00195 -0.00230 -0.00060 0.00460 -0.00308 0.00790 0.797 0.00195 0.00230 0.00080 -0.00140 -0.00269 0.00080 0.637 0.00195 0.00390 0.00240 0.00006 -0.00250 -0.00250 0.511 0.00195 0.00429 0.00289 0.00070 -0.00230 -0.00390 0.409 0.00195 0.00400 0.00281 0.00070 -0.00220 -0.00410 0.000 0.00195 0.00195 0.00195 0.00195 0.00195 0.00195 -0.409 0.00195 0.00400 0.00320 0.00099 -0.00201 -0.00339 -0.511 0.00195 0.00441 0.00330 0.00150 -0.00250 -0.00330 -0.637 0.00195 0.00419 0.00300 0.00039 -0.00310 0.00039 -0.797 0.00195 0.00330 0.00179 -0.00090 -0.00330 0.00250 -1.000 0.00195 -0.00029 -0.00179 -0.00330 -0.00320 0.00460 CHAPTER 10 BAR-LINKAGE FUNCTION GENERATORS WITH TWO DEGREES OF FREEDOM The preceding chapter has described a technique for the design of bar-linkage multipliers — a technique which is also applicable in the case of generators of arbitrary functions -with ideal grid structure. The present chapter will describe and illustrate a parallel technique for the design of bar linkages that generate a given function with nonideal grid structure. As in the preceding discussion, attention will be restricted to the use of the star grid generator. 10-1. Summary of the Design Procedure. — In designing a star grid generator for a multiplier, we began by considering an intersection nomogram for the given function, x3 = Xix2) (1) in the form of an ideal grid structure. We would have liked to carry out a topological transformation of this into an equivalent nomogram in which each family of lines would be a family of identical circles. The transformed nomogram would necessarily retain the ideal grid structure of the original one ; the corresponding star linkage would then be an ideal grid generator. We saw that such a transformation can not be found, but, guided by this idea, we succeeded in laying out three families of identical circles which had nearly the desired characteristics within a restricted region. Then, using graphical methods in adjusting the con- stants of the corresponding star linkage, we were able to make the grid structure take on more and more nearly the desired ideal form. The same line of thought can be followed in designing a star grid generator for an arbitrary function. Differences in the procedure arise principally from the fact that in improving the initial grid structure we cannot concentrate simply on making it ideal, but must at each step take account of the special function that is to be mechanized. To design a star grid generator for a given function of two independent variables, xs = f(xh a*), (2) we first represent it by an intersection nomogram consisting of three families of lines; 284 Sec. 10-1] SUMMARY OF THE DESIGN PROCEDURE 285 xi = x[r), (3a) x2 = x2*\ (36) x3 = xf. (3c) We desire to apply to this nomogram a topological transformation that will transform each family of curves into a family of circles of constant radius : 1. Circles C[r) of radius L\, centers A({\ on the line of centers Ci. 2. Circles C2a) of radius L2, centers A(2S), on the line of centers C2. 3. Circles Cf of radius L3, centers A(3t}, on the line of centers C3. If we can find such a transformed nomogram it will be equivalent to the original one, and from it we shall be able to determine the constants of the desired star grid generator: link lengths Lh L2, L%\ guiding curves Cij C2, Cz; scales with the x[r)-, x(2s)~, ^-calibrations at points A^\ A28\ Afy respectively. In seeking such a transformation we can be guided by the special characteristics, and especially the singularities, of the original nomo- gram. Under a topological transformation, intersections transform into intersections and points of tangency into points of tangency; these features of the original nomogram must then appear in the transformed nomogram. If all lines of constant Xi intersect at one point of the original nomogram, then all circles C\r) of the transformed nomogram must also intersect in a similar manner. If a line Xi = x\r) is tangent to the line x2 = x(2s) where Xz = x3l), then the circles C(xr) and C2S) must be tangent to each other at a point on the circle C3°. As a first step, we lay down tentative transforms of two of the original families of lines, let us say the xv and £2-contours. These will be families of circles that have the invariant characteristics of the original X\- and z2-contours, at least within the domain of mechanization; to each will be assigned a tentative value of X\ or x2, with due regard for all invariant characteristics of the original assignment. The tentative choice of these transforms is sufficient to determine the form of a tentative topological transformation and a corresponding new form of the third family of curves; to determine these curves we have only to plot, in the curvilinear coordinate system formed by the X\- and #2-circles, the curves Xz — xf as given by Eq. (2). If this tentative transformation should be the desired one, these rc3-contours will then be circles with the same radius. Of course, we cannot expect so fortunate a result. Usually, however, we can see how to rearrange or renumber the Xy and z2-circles so as to make the ^-contours roughly circular. We thus obtain a transformation of the original nomogram in which two of the families of contours have the desired circular form, and the third family has approximately the desired character. On replacing the third family of curves by a system of approximating circles with the same radius, we 286 BAR-LINKAGE FUNCTION GENERATORS [Sec. 10-2 can now convert it into the grid structure of a star linkage which gener- ates, at least approximately, the given function. It remains to modify this star linkage in such a way as to increase the precision with which it generates the given function, and at the same time to bring its scales into some convenient form, preferably circular. This can usually be accomplished by a method of successive approxi- mations. Accepting the x2- and z3-circles, C2S) and Cf, as determined above, we can replot the contours Xi = x[T) as denned by Eq. (2). These will be nearly circular; they can be approximated by a new system of circles C{r), which can often be so chosen that the line of centers has a simple and convenient form. The resulting star-linkage nomogram is usually more accurate than the first approximate nomogram, and more conveniently mechanized. Next, accepting the C(xr)- and C^-circles thus obtained, we can replot the contours x2 = x{.f and replace them by a new set of circles C(2S), and so on. There is no guarantee that this method will converge on a satisfactory solution of the problem; if it does not, the process must be begun again with a drastically different initial structure. An alternative method for improving the grid generator will be illustrated in Sec. 10-3. When a graphical method is no longer adequate for the further improvement of the grid structure, analytical methods can be brought into play. These, again, are essentially the same as those used in the design of multipliers. The grid generator, considered either separately or in combination with transformer linkages, generates a relation between the Xi-, x2-, ^3-scale readings which may be written as xz = f(xi, x2) + &(xi, x2). (4) The structural error &(xi, x2) is a function of all dimensions of the mecha- nism, as well as of x\ and x2. It can be evaluated for spectral values x[r) and x[s) of these latter variables and brought within specified tolerances by the methods described in Sec. 9-9. There is, however, no advantage in making a special choice of the spectra x[r) and x(2s\ except as this may be indicated by singularities or invariants of the given function. 10-2. Example: First Approximate Mechanization of the Ballistic Function in Vacuum. — As an example, we shall design a star grid gener- ator for the ballistic function in vacuum. The elevation angle xz of a gun that is to send a projectile through a point at ground range xh relative altitude x2, may be obtained by solving xi sin Xz cos xz — x2 cos2 x3 = -2 x\, (5) Sec. 10-2] THE BALLISTIC FUNCTION IN VACUUM 287 where v is the initial velocity of the shell and g is the acceleration of gravity. We shall take x\ and x2 as input variables in the linkage and generate the required elevation of the gun, x3, as the output variable. For simplicity in the calculation we shall take v = 500 m/sec and g = 10 m/sec.2 The parabolic trajectories of the shells then have the forms shown in Fig. 10*1. It will be noted that for each target position within the bounding envelope there are two possible values of x$. The larger value of x3 corresponds to a high trajectory, which becomes tangent to the envelope of the trajectories before the target is reached. The smaller value of x3 gives a lower trajectory, with shorter time of flight to the target; the shell reaches the target before it reaches the envelope of trajectories. 13,000 5000 20,000 25,000 10,000 15,000 Horizontal range xl in meters Fig. 10-1. — Trajectories of shells in vacuum. The dashed ellipse is the locus of maximum altitudes. These altitudes are indicated for some of the trajectories. We shall be interested only in the smaller of the two possible values of x3 and, correspondingly, only in the portions of the trajectories between the origin and the envelope. We shall also exclude from consideration the region of very small slant range, in the neighborhood of the singular point xi = Xi = 0, where xz is not defined. It is to be emphasized that the present example is intended only to illustrate a general technique and does not necessarily constitute the best solution of the stated problem. Equation (5), which is of relatively simple analytic form, can be mechanized by a net of standard com- puting mechanisms. Such a device will be less desirable mechanically than a bar-linkage function generator, but it will be much easier to design and free from structural errors. These advantages of a computing net are lost in the case of real ballistic functions, which offer no greater difficulties in bar-linkage design than does the present problem. In such 288 BAR-LINKAGE FUNCTION GENERATORS [Sec. 102 cases however, the usual practice is to separate the function to be gener- ated into two parts, one of simple analytical form, to be generated by a computing net, and the other a residue, to be generated by a bar linkage. The resulting problem in bar-linkage design will then usually be less difficult than that here considered, and the complete solution can be given much greater accuracy. The present example, nevertheless, offers a number of interesting points for discussion. Since the trajectories in Fig. 10-1 are contours of constant xz, the figure is actually an intersection nomogram that could serve for the solution of Eq. (5). We shall take it as the starting point of the design process and attempt to find a topological transformation that will transform the parabolas into a family of identical circles, the horizontal lines into a second family of identical circles, and the vertical lines into a third. Determination of the Xz-scale. — -The most difficult stage of the work is always the beginning; every possible clue must be used as a guide. We observe first that all the parabolas intersect in a common point, X\ = x2 = 0. If the desired transformation exists, it must carry these parabolas into a family of circles, of radius L3, which also intersect in a common point, the origin of the transformed nomogram. The centers of these circles must then he on a circle of radius L3 about the origin; this is the z3-scale, thus determined as to form and position, but having no known calibration points. One calibration point can of course be chosen at will, without loss of generality. We therefore begin construc- tion of the transformed nomogram, Fig. 10-2, by drawing the rt3-scale and the circle z3 = 0 with arbitrarily chosen radius L3; the calibration point xz = 0 on the z3-scale has been chosen to lie directly below the origin. Next we observe in Fig. 10-1 that the contour x2 = 0 is tangent to the trajectory z3 = 0 at the origin and lies above it everywhere else. The transformed circle x2 = 0 must then be tangent to the transformed circle x3 = 0 at the origin; in Fig. 10-2 its center must lie directly above or directly below the origin. Comparison with Fig. 10-1 suggests that its center should lie below the origin, and that its radius, L2, should be greater than L3. It has been so drawn in Fig. 10-2. The choice of L2, which has been made arbitrarily, can be changed at will if the design procedure should fail to progress satisfactorily. This choice of the circle x2 = 0 also fixes the point x2 = 0 on the x2-scale at a distance L2 below the origin. Guided by the distribution of intersections of the trajectories xz = xf with the horizontal line x2 = 0, we are now in a position to make a tentative calibration of the z3-scale. It is a familiar fact that projectiles shot in vacuum at elevation angles x$ and 90° — x$ will have the same Sec. 10-2] THE BALLISTIC FUNCTION IN VACUUM 289 horizontal range; for instance, the parabolas Xz = 40° and xz = 50° of Fig. 10-1 intersect on the line x2 = 0. The trajectory xz = 45° gives the greatest horizontal range. The transformed circle Xz = 45° in Fig. 10-2 must then intersect the circle x2 = 0 at the greatest possible distance from the origin — a distance equal to the diameter of the ^-circles. This intersection point, S in Fig. 10-2, can be determined by use of a compass. The midpoint of the diameter OS lies on the Xz scale at the calibration point xz = 45°; calibration points for xz < 45° will lie below this point, calibration points for Xz > 45°, above it. Fig. 10-2. -Construction of tentative scales in mechanization of the ballistic function in vacuum. We shall now choose other calibration points on the z3-scale, such that the circles x3 = xf and Xz = 90° — xf intersect the circle x2 = 0 at the same point. Specifically, having chosen calibration points for xz = 10°, 20°, 30°, 40°, which interpolate between the known points xz = 0° and x3 = 45°, we shall construct the corresponding circles of the transformed nomogram. Through the intersection of each such circle with the line x2 = 0, and the origin, we shall construct a circle of radius La. These circles correspond to x3 = 80°, 70°, 60°, 50°, respectively; their centers are the desired calibration points on the £3-scale. In pro- ceeding thus, we are guided by properties of descending portions of the 290 BAR-LINKAGE FUNCTION GENERATORS [Sec. 10-2 parabolic trajectories with x3 > 45°, which have no immediate relevance to the chosen problem. . The relations that we have established are thus useful guides in the preliminary calibration of the £3-scale, but they need not be maintained throughout later developments. First of all, we note that the circle Xz = 90°, like the associated circle Xi = 0°, must be tangent to the circle x2 = 0; its center must lie directly above the origin. The z3-scale must then extend through an arc of 180°. Since the scale from Xz = 0° to Xz = 45° covers less than 90° of this arc, the spacing between consecutive ^-calibrations must, on the average, increase with Xz) we can reasonably assume that this increase continues smoothly throughout the length of the scale. We shall therefore choose calibration points Xz = 10°, 20°, 30°, 40°, which have gradually increasing separations and which lead to the determination of points x3 = 50°, 60°, 70°, 80°, with separations that fall into the same smoothly increasing sequence. Such points are easily found; they are shown in Fig. 10-2, together with the associated circles of radius L3. Determination of the x2-scale. — Thus far we have established only the point x2 = 0 on the z2-scale. As our principal clue in the further con- struction of this scale we have the points of tangency of the x2- and z3-contours. In Fig. 10-1 the horizontal line x2 = 375 is tangent to the parabola x3 = 10° at its vertex; the transformed circle x2 = 375 in Fig. 10-2 should similarly be tangent to the transformed circle Xz = 10°. The circle x3 = 10° has already been determined, but we know only the radius of the circle x2 = 375; its center — the point x2 = 375 on the x2- scale — may lie anywhere on a circle of radius L2 — L3, with its center at the point Xz = 10°. An arc of this circle near the point x2 = 0 is shown in Fig. 10-2. Similarly, the scale points x2 = 1460, 3125, 5160, 7335, 9370, 11040, 12130, 12500, must lie on circles of radius L2 - L3 about centers on the x3-scale, at the points xz = 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°, respectively. Arcs of these circles also appear in Fig. 10-2. There is another clue to the nature of the £2-scale, but it is relatively unreliable. It is well known that the points of maximum x2 on the trajectories lie on an ellipse (the dashed curve of Fig. 10-1). These maxima occur at first close to the origin 0; their £i-coordinates increase with Xz, and then decrease to 0 as Xz goes to 90°. Now, the transformed nomogram under construction bears some similarity to the original one, Fig. 10-1, in which x2 increases from bottom to top, X\ from left to right. In view of this we may expect the points of tangency between the trans- formed x2- and z3-circles to move at first toward the right, as x3 increases, and then as far as possible to the left. If this is to be the case, the X2-scale must then rise to the right of the origin, with its upper end E, corresponding to x2 = 12500, at about the same level as the scale point Xz — 90°. Figure 10-2 shows such a choice of E. Sec. 10-2] THE BALLISTIC FUNCTION IN VACUUM 291 A preliminary choice of the z2-scale can now be made. For mechan- ical reasons it has been constructed in Fig. 10-2 as a circular arc— a tentative choice that can be modified at any time. The calibrations on this scale are determined by its intersections with the arcs already con- structed. For later use, calibration points have been interpolated for evenly spaced values of x2: 1000, 2000, 3000, . . . 12000. These are easily and accurately determined by plotting on cross-section paper a smooth curve of distance along the £2-scale against the value of x2, using the known calibration points, and reading off the distances corresponding to the chosen values of x2. This completes the determination of our tentative topological trans- formation. Contours of constant x2 can be drawn in at will, but are omitted from Fig. 10-2. Determination of the Xi-scale. — We have next to construct contours of constant Xi in the transformed nomogram. This is conveniently done using computed values of x2 for a series of values of xi and xh as given in Table 10-1. Only values within our restricted domain of interest are tabulated. Table 10-1. — Values of x2 Computed by use of Eq. (5) Xz ^\ 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 80° 8650 12050 70° 4830 8250 10300 11030 10350 60° 3150 5650 7500 8700 9350 9250 50° 2150 4000 5400 6450 7100 7330 7200 6700 5750 4500 40° 1550 2800 3800 4550 5000 5150 5050 4700 4050 3150 30° 1150 1900 2500 2900 3100 3100 2830 2400 1730 850 20° 650 1100 1350 1450 1350 1100 600 0 10° 280 375 300 75 To construct the curve x\ = 2000 we refer to the first column of Table 10-1. We construct arcs of radius L2 with centers at the points x2 = 8650, 4830, 3150, 2150, 1550, 1150, 650, 280, intersecting, respec- tively, the circles xz = 80°, 70°, 60°, 50°, 40°, 30°, 20°, 10°. The points of intersection lie on the curve X\ = 2000. In the same way we can deter- mine points on the other contours of constant x2, as shown in Fig. 10-2. Because of irregularities in the x2- and rr3-scales, these points do not lie on smooth curves, but on rather irregular ones; they have been connected by straight lines in Fig. 10-2, merely to bring out their relations. It would have been very gratifying if the contours of constant Xi had turned out to be circles of constant radius L\. The actual result is not bad, for a first trial, since the curves do resemble arcs of circles. The radii of these circles are not exactly equal, but it is not difficult to select an average radius L\. 292 BAR-LINKAGE FUNCTION GENERATORS [Sec. 10-3 We have now to construct the Zi-scale. About each of the established points of the contour Xi = 2000 we construct arcs of radius L\. These intersect near the upper margin of Fig. 10-2 and thus determine roughly the position of the point X\ = 2000 on the zi-scale. Similar construc- tions are shown for the established points of the other Xi-contours. The intersections are rather diffuse; the form of the Zi-scale is not determined very precisely. Fortunately, the most diffuse intersections occur for the least critical part of the Zi-scale, the center. For these values of xi, the x2- and rr3-contours are very nearly tangent to each other, and the computed value of Xz is very insensitive to the value of X\. For instance, let us consider a case in which the central pivot of the star linkage is at a point of tangency of the x2- and z3-contours. As long as x2 is fixed, any displacement of the Zi-input — whether along or perpendicular to the scale — can move the star pivot only along the z2-circle and thus produce at most a second-order change in x$. We have, therefore, to attach little importance to the diffuseness of the intersection in the central part of the £j-scale: we can adjust the position of that part of the scale and its calibration points with relative freedom. The reason for this is also apparent on inspection of Fig. 10-1; for values of x\ near 12000, a change in X\ with constant x2 carries one very nearly along a trajectory with constant xz. (The steeply descending trajectories we have already excluded from consideration.) It is evident that the £i-scale will be nearly circular; in Fig. 10-2 it has been given an accurately circular form. It then becomes clear that the calibrations in X\ will be almost equally spaced. This fact suggests that an exactly even scale in X\ should be laid down — a procedure that has been followed in Fig. 10-2. We have thus given to the £i-scale a par- ticularly simple form, which we can hope to maintain through later stages of the development. 10-3. Example : Improving the Mechanization of the Ballistic Func- tion in Vacuum. — In following the method described in Sec. 10-1 for the improvement of our preliminary mechanization of the ballistic function, we can accept the x%- and z3-scales already defined and reconstruct the £2-scale. It should now be sufficiently clear how this work would proceed. We shall therefore apply another useful technique to this problem. Let us accept the very convenient Zi-scale of Fig. 10-2 and the estab- lished values of Lh L2, and L3. Instead of prescribing the avscale directly, we shall keep unchanged the contour x3 = 40°, and shall require that the new linkage give an exact solution of the problem whenever xz = 40°. This requirement will completely define the rr2-scale for all values of x2 less than 5160. For instance, we can locate the scale point x2 = 1000 in the following manner. From the original nomogram we read that when xz = 40° and x2 = 1000, x1 may be 1300 or 23300. About Sec. 10-3] IMPROVING THE MECHANIZATION 293 the points X\ = 1300 and Xi = 23300 on the #i-scale we draw circular arcs of radius Lh intersecting the contour xz = 40° at points A and B (Fig. 10-3). These points must both lie on the contour x2 = 1000; we therefore construct about them arcs of radius L2, and locate the scale point X2 = 1000 at their intersection. The scale points x2 = 2000, 3000, 4000, 5000, can be determined similarly by the use of the data in Columns 1 and 2 of Table 10-2. Table 10-2 . — Values of xi Computed by Eq. (5) "\^^ x3 Xi ^^^ 40° 50° 60° 70° 80° 1000 1300 23320 2000 2700 21900 3000 4350 20250 2850 4000 6450 18150 4000 2600 1600 800 5000 10000 14500 5300 3400 2100 1000 6000 7000 17600 4350 2600 1240 7000 9600 14950 5400 3160 1500 8000 6700 14950 3850 1800 9000 8650 12950 4600 2100 10000 5600 10450 2480 11000 7600 8400 2950 12000 3850 The scale points thus established lie on a circular arc with center Q2 (Fig. 10-3) and are equidistant to within the accuracy of the construction. We shall therefore complete the a;2-scale by extending it as a circular arc, with equidistant scale divisions up to x2 = 12000. It remains to reconstruct the contours of constant xs and the £3-scale. Points on the contours x3 = 10°, 20°, 30° are conveniently located by the use of the data in the last three rows of Table 10-1; they lie at the intersections of arcs of radius Li and L2 about corresponding points on the Xi- and ^2-scales, respectively. Points on the contours Xz = 50°, 60°, 70°, 80° can be located similarly by the use of data given in Table 10-2. All these points are shown in Fig. 10-3 as small circles. Through them we can pass circles of radius Lz, with errors that are appreciable only at the outer extremity of a few of the arcs. The centers of these circles lie on a nonuniform x3-scale that is only roughly circular. The fact that the Xy and #2-scales are even and circular makes this solution of the problem attractive, even though the #3-scale is of the most general type. No transformer linkages will be required for the inputs ; it remains to design a single bar linkage that will both guide the £3-point of the star linkage over the present noncircular #3-scale and provide an out- put motion linear in x5. How this can be done will be shown in Sec. 10-4. Figure 10-4 shows a schematic layout of the linkage in its present state, 294 BAR-LINKAGE FUNCTION GENERATORS [Sec. 10-3 with a curved slide for the output terminal. The elevation scale is restricted to the range 10° < rr3 < 80°, which alone would be important if the mechanized ballistics had practical significance. This function gen- erator has a very small error for slant ranges greater than 2000 m, except for a few points close to the envelope of trajectories. (The solution near Xi = Xi = 0 is poor because no attempt was made to force the contours Fig. 10-3. — Construction of improved scales for mechanization of the ballistic function in vacuum. of constant rc3 to intersect at a single point.) This design could be improved somewhat by graphical methods, without sacrificing the simple forms of the xx- and z2-scales; still further improvement could be obtained by applying analytic methods. In practice, however, maximum accu- racy could be obtained by reformulation of the problem, introduction of new variables, and use of the bar linkage to mechanize a function of more suitable character. Sec. 104] CURVE TRACING AND TRANSFORMER LINKAGES 295 104. Curve Tracing and Transformer Linkages for Noncircular Scales. — Practical application of a grid generator with a nonuniformly curved scale requires solution of two problems: 1. Design of a constraint for the grid-generator terminal that is more satisfactory than a curved slide. 2. Design of a transformer linkage to provide a satisfactory external terminal, usually with an even scale. Horizontal range jc1=0 o 10,000 ol5,000 o o 0 20,000 Fig. 10-4. — Schematic layout of ballistic computer. These problems can often be solved simultaneously by a device such as that sketched in Fig. 10-5, in which the terminal T of the grid generator is pivoted to and guided by a rigid extension QTR of the central link of a three-bar linkage PQRS. One of the cranks of this linkage may itself serve as the terminal of the transformer, as shown in Fig. 10-5, or a harmonic transformer may be added to give a slide terminal, as in the multiplier of Fig. 8-16. We shall consider here the first and simpler alternative ; the procedure is easily extended to the second case by use of the ideas presented in Chap. 8. 296 BAR-LINKAGE FUNCTION GENERATORS [Sec. 104 Let us consider that the rigid triangle QTR of Fig. 10*5 consists of rigid bars. The device may then be divided into two parts : 1. A transformer linkage, consisting of the link TR and the crank RS. When the joint T is guided along the scale A B, these elements serve to transform readings on the uneven scale AB into identical read- ings on the even circular scale CD. 2. A constraint linkage, consisting of the crank PQ and the links QR and QT. This, together with the elements of the transformer linkage, guides the joint T along the scale AB. In designing such a linkage these parts are considered separately, and in the order listed. Fig. 10-5. — Linkage transformer from an uneven noncircular scale AB to an even circular scale CD. The joint T will follow AB without further constraint. The Transformer Linkage. — The transformer linkage can be designed by application of the geometric method for three-bar-linkage design, as described in Chap. 5. To understand how this can be done without any significant change in the procedure, we may consider the three-bar linkage from a point of view not previously emphasized. A three-bar linkage can be used to generate a relation between an input variable xi, which can be read on a uniform scale associated with the input crank, and an output variable x2, which can be read on a uniform scale associated with the output crank. Now, a nonlinear mechanization of the same relation can be obtained by use of the output crank alone; the uniform X2-scale may be supplemented by a nonuniform £i-scale, and the output crank used simply as a pointer to indicate corresponding values of Xi and x2. The function of the input crank and connecting link in the three-bar linkage is, then, that of transforming this circular but nonuniform scale of x\ into a uniform circular input scale. It will be observed that the geometric method of three-bar-linkage design can be understood from this point of view; it will be noted also that the circular form of the output scale plays no essential Sec. 104] CURVE TRACING AND TRANSFORMER LINKAGES 297 role in the procedure. This method can be applied, then, whenever it is desired to carry out a transformation between a circular scale and another scale of arbitrary form. (Interchange in the roles of terminals as input and output does not affect the procedure.) For example, it can be used in the design of highly nonideal harmonic transformers, in which one scale is linear, as well as in the present case, where one scale may have arbitrary form. To design the transformer linkage one can then proceed as follows: 1. Choose a spectrum of values of the variable (for example, x) with differences that change in geometric progression, with ratio g: g — 1 2. Construct the given scale A B, and on it lay down the points cor- responding to these spectral values of x(r). These points cor- respond to the point P(0), P(1), . . . PM, of Fig. 5-21. 3. About the points P(r) construct circles C(r) with radius B2 = TR. This completes a chart corresponding to Fig. 5*21. 4. Over this chart move the characteristic overlay of the geometric method, Fig. 5*23, constructed for the chosen value of g. If it is possible to find a position of this chart (face up or face down) in which successive circles C(0), C(1), C(2), . . . pass through the inter- sections of an overlay circle with successive radial lines, the required constants of the transformer linkage can be read off at once. The position of the pivot S is the center of the overlay. The circular scale CD must coincide with the overlay circle on which the fit was found; the calibration points on this scale corresponding to the chosen spectrum x(r) lie at the intersection with this circle of the circles C(r). The length of the connecting link TR corresponds to the arbitrarily chosen length B2. 5. If necessary, try a succession of values of B2 in order to find that which gives the best fit. Figure 10-6 shows a transformer linkage, thus designed, for the rr3-scale of Fig. 10-4. The Constraint Linkage. — The possible paths of the joint Tin linkages of this type are the three-bar curves discussed, from a more mathematical point of view, by Roberts, Cayley, and Hippisley.1 Even as restricted by the choice of the elements TR and RS of the transformer linkage, this 1 S. Roberts, "On Three-Bar Motion in Plane Space," Proc. Math. Soc, Lond., 7, 14 (1875). A. Cayley, "On Three-bar Motion," Proc. Math. Soc, Lond., 7, 136 (1875). R. L. Hippisley, "A New Method of Describing a Three-bar Curve," Proc. Math. Soc, Lond., 18, 136 (1918). 298 BAR-LINKAGE FUNCTION GENERATORS [Sec. 10-4 is a large family of curves, with which a great variety of curves AB can be fitted accurately. A simple graphical method suffices for the design of these linkages. We wish to choose lengths for the bars TQ and QR such that when the points T and R move over their respective scales the point Q will describe a circle. If we then constrain Q to move on this circle, by means of the crank PQ, and R to move on the circle R, by means of the crank SR, the joint T will be constrained to move along the scale AB, as desired. We therefore prepare a chart that shows the scales AB and CD in their proper relation. Over this chart we place a transparent overlay on which is marked a line of length B2) representing the bar TR. If we now Fig. 10-6. — Transformer linkage for the X3-scale of Fig. 10-4. move the ends T and R of this bar along their respective scales, the other points of the overlay will have the motions of points rigidly attached to the bar TR. The path traversed by any point of the overlay can quickly be laid out on the chart. Comparison of a number of these paths will usually call attention to a region on the overlay — in addition to that near the point R — that traverses a nearly circular path. Comparison of the paths of a few points of this region will then suffice for the location on the overlay of the point Q that has the most nearly circular path. The length of the bars TQ and QR can then be measured on the overlay; the pivot P will be located at the center of the circular path, and PQ will have a length equal to its radius. This will complete the determination of the linkage constants — to the accuracy possible by graphical methods. Figure 107 shows, for the example of the preceding sections, the paths of a number of points of the overlay. Since the point T moves over a roughly circular path AB, it was to be expected that a very nearly circular path, QQ' , would be found near by — that the bar length TQ would be small. A sketch of the completed transformer-and-constraint linkage for this example is shown in Fig. 10-8. Sec. 10-41 CURVE TRACING AND TRANSFORMER LINKAGES 299 Fig. 10-7. — Paths of points rigidly attached to the bar TR, as its ends move along scales AB and CD. The point P is the center of the circular path QQ'. The scales are the same as in Fig. 10-6. Fig. 10-8. — Complete transformer-and-constraint linkage for Z3-scale of Fig. 10-4. APPENDIX A TABLES OF HARMONIC TRANSFORMER FUNCTIONS An extended discussion of the structure and use of these tables will be found in Sees. 4-3 to 4-7. A'l. Table A-l. Hk as a Function of 0t. — Each unit of this table may be read in two ways, according to the following schemes: 9 Si Hk El 1 - - Bi 1 - - Bi Ht Hk Bi 9 Xim XiM The defining relations are tt ._ sin (Xim + BjAXj) — (sin X{)min (sin X»)max — (sin Xi)min 9 = cos {Xim + BjAXj) - (cos Xi)1 (COS Xt)max — (COS Xi),^ (cos Xi)mai — (cos Xt)min (sin Xi)t (sin Xi)i (4-12) (4-42) (4-31) where max and min indicate, respectively, the maximum and minimum values of the function in question, when Xim = Xi ^ XiM = Xim ~h AJi. A*2. Table A«2. Bi as a Function of H k. — In each column are tabu- lated values of Bi corresponding to equally spaced values of Hk, for the values of Xim and XiM shown at the top of the column. 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O05hNCW00OM(NO O00O(NCCOh(NhNO OO(NMTfiON00OCO §8 OS ©OOOOOOOOOO ooooooooooo OOOONOiO^MINhO ooooooooooo to O © OSIOOOOOCOOlOOSOO ■*OcOONhiOW^(NO C0cO00OSOSO00t>tOC0O O 0 oo H tO ooooooooooo O 0 OO CO 00 1 OCOcOOCOt>l>-COb-tOO OOIhhOhnoMhO ONOCOMOOOOOiCO OOoCOOlocOOOOOOsO 00 OS co OO OO ooooooooooo OoCOCOOlocOI>000© ooooooooooo OOS001>COtOOCOCOoO ooooooooooo OS to OS CO © OOSCOt>-oottocOOOsO oooOCOCOooCOooo (NONOJOOOKNOONOO (Not^GOOSOSOt^tOCOO o o tO tO ^H IO o'oo'do'ddcJo'oo 0 0 tO to cot> 1 OOcOi*iOOOOMNO OOSCNJtOOOOOiOO^HO OOONOCOiONNiOO OOHfq^iOcDNOOOO O CO CO to OO ooooooooooo Oo(NCOOiocOI>OOOsO ooooooooooo OOSOOOcOtOOCOCOoO OOOOOOOOOOO O to CO © OocOojiOOSocOoO ooaw^o^iNooo OCOOOOtOOtOCOOCOO ococoooosoosoococoo 0 0 OO CO CO ooooooooooo 0 0 OO 1 OOONMONOOOHO OOO^OOOOOOSCOOO OtOOiot^OCMOio^O ©©ocOCOtOcOt^OOOsO CO O to 00 OO ooooooooooo «b Oo(NCOTjHtOcOI>OOOsO ooooooooooo tO tO coo oo o o O©00N©iO^C0(NhO ooooooooooo o o o o i-H oi>oosOsoasCOCOCOO COOOOScOCSOStOOCOCOO 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00 OsO O OO OO Oo O o 00 d NHiONtONOCOtOOO OocsCOOiOOOOoOt^O OI>O00C0C0OC0tOI>O OsOsOSOSOsOOcOtocOoO o O OO oo oo 1 *- OOOOOOOOOOO 0 0 OO CO o 1 * OcOoCscOOsI>tOtOCOOO OOsOoiOoCNJoCOOO O^^N(NCONiOCSQ>0 OoCOocOI>OOOsOSOO I— 1 OOOOOOOOOOO 0 0(MCOO»OCDI>00 050 ooooooooooo OOS001>cOiOocOtMoO ooooooooooo 324 APPENDIX A Table A-2. — 0* as a Function of Hk AX, = 40° • Hk Xim = -90° -85° -80° -75° -70° -65° -60° -55° -50° -45° XiM = -50° -45° -40° -35° -30° -25° -20° -15° -10° -5° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.3104 0.2424 0.2021 0.1769 0.1600 0.1478 0.1387 0.1315 0.1256 0.1206 0.2 0.4399 0.3805 0.3398 0.3111 0.2899 0.2736 0.2607 0.2500 0.2410 0.2332 0.3 0.5398 0.4892 0.4523 0.4245 0.4030 0.3858 0.3715 0.3595 0.3490 0.3398 0.4 0.6246 0.5822 0.5501 0.5251 0.5050 0.4884 0.4743 0.4622 0.4514 0.4417 0.5 0.6998 0.6652 0.6383 0.6167 0.5990 0.5840 0.5710 0.5596 0.5494 0.5400 0.6 0.7682 0.7410 0.7194 0.7018 0.6870 0.6742 0.6630 0.6530 0.6439 0.6355 0.7 0.8314 0.8114 0.7952 0.7817 0.7702 0.7602 0.7513 0.7432 0.7357 0.7287 0.8 0.8906 0.8775 0.8667 0.8576 0.8497 0.8428 0.8365 0.8308 0.8254 0.8202 0.9 0.9467 0.9402 0.9348 0.9302 0.9262 0.9225 0.9192 0.9162 0.9133 0.9106 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk Xim = -40° -35° -30° -25° -20° -15° -10° i -5° 0° 5° XiM = 0° 5° 10° 15° 20° 25° 30° 35° 40° 45° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1164 0. 1126 0.1091J0. 1060 0.1030 0.1002 0.0975 0.0948 0.0921 0.0894 0.2 0.2264 0.2201 0.2144 0.20900.2040 0.1990 0.1942 0.1894 0.1846 0.1798 0.3 0.3315 0.3238 0.3167 0.3099 0.3034 0.2970 0.2908 0.2844 0.2780 0.2713 0.4 0.4328 0.4246 0.4168 0.4093 0.4020 0.3947 0.3874 0.3801 0.3725 0.3645 0.5 0.5313 0.5231 0.5152 0.5076 0.5000 0.4924 0.4848 0.4769 0.4687 0.4600 0.6 0 . 6275 0.6199 0.6126 0.6053 0.5980 0.5907 0.5832 0.5754 0.5672 0.5583 0.7 0.7220 0.7156 0.7092 0.7030 0.6966 0.6901 0.6833 0.6762 0.6685 0.6602 0.8 0.8154 0.8106 0.8058 0.8010 0.7960 0.7910 0.7856 0.7799 0.7736 0.7668 0.9 0.9079 0.9052 0.9025 0.8998 0.8970 0.8940 0.8909 0.8874 0.8836 0.8794 1.0 1.0000 1 . 0000 1 . 0000 1 . 0000 1 . 0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk 0.0 Xim = 10° 15° 20° 25° 30° 35° 40° 45° 50° XiM = 50° 55° 60° 65° 70° 75° 80° 85° 90° 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0867 0 . 0838 0 . 0808 0 . 0775 0 . 0738 0.0698 0.0652 0.0598 0.0533 0.2 0.1746 0.1692 0.1635 0.1572 0.1503 0.1424 0.1333 0.1225 0.1094 0.3 0.2643 0.2568 0.2487 0.2398 0.2298 0.2183 0.2048 0.1886 0.1686 0.4 0.3561 0.3470 0.3370 0.3258 0.3130 0.2982 0.2806 0.2590 0.2318 0.5 0.4506 0.4404 0.4290 0.41600.4010 0.3833 0.3617 0.3348 0.3002 0.6 0.5486 0.5378 0.5257 0.51160.4950 0.4749 0.4499 0.4178 0.3754 0.7 0.6510 0.6405 0.6285 0.6142 0.5970 0.5755 0.5477 0.5108 0.4602 0.8 0.7590 0.7500 0.7393 0.7264 0.7101 0.6889 0.6602 0.6195 0.5601 0.9 0.8744 0.8685 0.8613 0.8522 0.8400 0.8231 0.7979 0.7576 0.6896 1.0 1.0000 1.0000 1.0000 1.00001.0000 1.0000 1.0000 1.0000 1.0000 APPENDIX A Table A -2. — 0* as a Function of Hk — (Cont.) 325 AXi = 50° Hk Xim = -90° XiM = -40° -85° -35° -80° -30° -75° -25° -70° -20° -65° -15° -60° -10° -55° -5° -50° 0° 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0000 0.3072 0.4358 0.5354 0.6201 0.6955 0.7643 0.8283 0.8884 0.9455 1.0000 0.0000 0.2492 0.3853 0.4922 0.5839 0.6658 0.7409 0.8109 0.8769 0.9397 1.0000 0.0000 0.2111 0.3479 0.4585 0.5546 0.6413 0.7212 0.7960 0.8669 0.9347 1.0000 0.0000 0.1855 0.3198 0.4318 0.5306 0.6206 0.7043 0.7831 0.8582 0.9303 1.0000 0.0000 0:1673 0.2980 0.4100 0.5105 0.6029 0.6895 0.7716 0.8503 0.9262 1.0000 0.0000 0.1539 0.2806 0.3919 0.4932 0.5874 0.6763 0.7612 0.8430 0.9224 1.0000 0.0000 0.1435 0.2664 0.3766 0.4782 0.5736 0.6644 0.7516 0.8363 0.9189 1.0000 0.0000 0.1352 0.2544 0.3632 0.4648 0.5611 0.6534 0.7427 0.8339 0.9154 1.0000 0.0000 0.1283 0.2441 0.3515 0.4527 0.5496 0.6431 0.7343 0.8237 0.9121 1.0000 Hi 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xim = -45c XiM = 5C 0.0000 0.1224 0.2350 0.3408 0.4416 0.5388 0.6334 0.7262 0.8178 0.9089 1 . 0000 40° 10° 0.0000 0.1173 0.2269 0.3311 0.4313 0.5287 0.6241 0.7182 0.8119 0.9056 1 . 0000 35° 15° 0000 1127 2195 3221 4215 5189 6150 7104 8060 9022 0000 30° 20° 0000 1086 2127 3135 4122 5094 6060 7026 8000 0000 ■25° 25° 0000 1048 2062 3054 4030 5000 5970 0 6946 0 20° 30° 7938 8952 0000 0000 1012 2000 2974 3940 4906 5878 6865 7873 8914 0000 ■15° 35° 0.0000 10° 40° 0978 1940 2896 3850 4811 5785 6779 7805 8873 0000 0.0000 -5C 45c 0944 1881 2818 3759 4713 5687 6689 7731 8827 0000 .0000 .0911 .1822 .2738 .3666 .4612 .5584 .6592 .7650 .8776 .0000 Hk xim = 0° 5° 10° 15° 20° 25° 30° 35° 40° XiM = 50° 55° 60° 65° 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0879 0.0846 0.0811 0.0776 0.0738 0.0697 0.0653 0.0603 0.0545 0.2 0.1763 0.1661 0.1637 0.1570 0.1497 0.1418 0.1331 0.1231 0.1116 0.3 0.2657 0.2573 0.2484 0.2388 0.2284 0.2169 0.2040 0.1891 0.1717 0.4 0.3569 0.3466 0.3356 0.3237 0.3105 0.2957 0.2788 0.2591 0.2357 0.5 0.4504 0.4389 0.4264 0.4126 0.3971 0.3794 0.3587 0.3342 0.3045 0.6 0.5473 0.5352 0.5218 0.5068 0.4895 0.4694 0.4454 0.4161 0.3799 0.7 0.6485 0.6368 0.6234 0.6081 0.5900 0.5682 0.5415 0.5078 0.4646 0.8 0.7559 0.7456 0.7336 0.7194 0.7020 0.6802 0.6521 0.6147 0.5642 0.9 0.8717 0.8648 0.8565 0.8461 0.8327 0.8145 0.7889 0.7508 0.6928 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 326 AXi = 60° APPENDIX A Table A-2. — 6i as a Function of Hk — (Cont.) Hi 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xim XiM 90 c 30c 0.0000 0.3032 0.4307 0.5298 0.6145 0.6902 0.7595 0.8243 0.8855 0.9439 1.0000 85* 25c -80c -20c 0.0000 0 0.2524 0 0.3863|o 0.4917:0 0.5823 0 0.6635 0 0.73840 0.80850 0.8750 0 0.9386 0 1.0000 1 ■75c ■15c 0000 0 2166 0 3517 4605 5551 6406 7199 7944 8655 93380 0000 1 0000 1910 3243 4347 5319 6206 7034 7817 8568 9293 0000 70° 10° 0000 1722 3024 4129 5118 6029 6885 7701 8487 9251 •65° -5° 60 c 0C 0000 0. 1579 0, 0000 1 I 2843 3943 4941 5869 6749 7592 8411 9211 0000 0000 1465 2691 3780 4782 5723 6622 7490 8338 9172 00001 -55c 5C 0000 1373 2561 3637 4638 5588 6503 7392 8267 9133 0000 -50c 10c .0000 .1295 .2447 .3507 .4506 .5462 .6389 .7297 .8197 .9095 .0000 Hk Xim = -45° -40° -35° -30° -25° -20° -15° -10° -5° XiM = 15° 20° 25° 30° 35° 40° 45° 50° 55° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1229 0.1170 0.1118 0.1070 0. 1026 0.0984'0. 0944 0.0905 0.0867 0.2 0.2346 0.2254 0.2169 0.2090 0.2015 0.1943 0.1873 0.1803 0.1733 0.3 0.3389 0.3279 0.3176 0.3077 0.2982 0.2888.0.2796 0.2703 0.2608 0.4 0.4382 0.4265 0.4153 0.4044 0.3936 0.3829 0.3721 0.3611 0.3497 0.5 0.5341 0.5225 0.5112J0.5000 0.4888 0.47750.4659 0.4538 0.4412 0.6 0.6279 0.6171 0.60640.5957 0.5847 0.5735 0.5618 0.5494 0.5362 0.7 0.7204 0.7112 0.7018 0.6923 0.6824 0.67210.66110.6493 0.6363 0.8 0.8127 0.8057 0.7985 0.7910 0.7831 0 . 7746!0 . 7654 0 . 7553 0 . 7439 0.9 0.9056 0.9016 0.8974 0.8930 0.8882 0 . 8830'0 . 8771 0 . 8705 0 . 8627 1.0 1.0000 1.0000 1.0000 1.0000 1 . 0000 1 . 0000| 1 . 0000, 1 . 0000, 1 . 0000 Hk Xim = 0° 5° 10° 15° 20° 25° 30° XiM = 60° 65° 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0828 0.0789 0.0749 0.0707 0.0662 0.0614 0.0561 0.2 0.1662 0.1589 0.1513 0.1432 0.1345 0.1250 0.1145 0.3 0.2510 0.2408 0.2299 0.2183 0.2056 0.1915 0.1757 0.4 0.3378 0.3251 0.3115 0.2966 0.2801 0.2616 0.2405 0.5 0.4277 0.4131 0.3971 0.3794 0.3594 0.3365 0.3098 0.6 0.5218 0.5059 0.4882 0.4681 0.4449 0.4177 0.3855 0.7 0.6220 0.6057 0.5871 0.5653 0.5395 0.5083 0.4702 0.8 0.7309 0.7157 0.6976 0.6757 0.6483 0.6137 0.5693 0.9 0.8535 0.8421 0.8278 0.8090 0.7834 0.7476 0.6968 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 APPENDIX A Table A-2. — 0» as a Function of Hk — (Cont.) 327 AXi = 70c Hk Xim = -90° -85° -80° -75° -70° -65° -60° -55° XiM = -20° -15° -10° -5° 0° 5° 10° 15° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2986 0.2530 0.2193 0.1942 0.1750 0.1600 0.1479 0.1379 0.2 0.4247 0.3847 0.3522 0.3257 0.3037 0.2851 0.2692 0.2553 0.3 0.5231 0.4885 0.4592 0.4341 0.4124 0.3934 0.3764 0.3611 0.4 0.6077 0.5782 0.5525 0.5299 0.5097 0.4915 0.4749 0.4596 0.5 0.6836 0.6591 0.6372 0.6175 0.5997 0.5832 0 . 5678 0.5533 0.6 0.7537 0.7339 0.7161 0.6997 0.6846 0.6704 0.6569 0.6439 0.7 0.8194 0.8045 0.7908 0.7781 0.7661 0.7547 0.7437 0.7328 0.8 0.8819 0.8718 0.8625 0.8537 0.8453 0.8371 0.8291 0.8211 0.9 0.9419 0.9368 0.9320 0.9274 0.9230 0.9186 0.9142 0.9098 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk Xim = -50° -45° -40° -35° -30° -25° -20° -15° XiM = 20° 25° 30° 35° 40° 45° 50° 55° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1294 0.1221 0.1156 0.1098 0.1045 0.0995 0.0947 0.0902 0.2 0.2430 0.2320 0.2218 0.2124 0.2036 0.1951 0.1869 0.1789 0.3 0.3472 0.3342 0.3221 0.3105 0.2994 0.2886 0.2778 0.2672 0.4 0.4452 0.4316 0.4186 0.4059 0.3935 0.3811 0.3687 0.3561 0.5 0.5394 0.5260 0.5129 0.5000 0.4871 0.4740 0.4606 0.4467 0.6 0.6313 0.6189 0.6065 0.5941 0.5814 0.5684 0.5548 0.5404 0.7 0.7222 0.7114 0.7006 0.6895 0.6779 0.6658 0.6528 0.6389 0.8 0.8131 0.8049 0.7964 0.7876 0.7782 0.7680 0.7570 0.7447 0.9 0.9053 0.9005 0.8955 0.8902 0.8844 0.8779 0.8706 0.8621 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk Xim 10° -5° 0° 5° 10° 15° 20° XiM = 60° 65° 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0858 0.0814 0.0770 0.0726 0.0680 0.0632 0.0581 0.2 0.1709 0.1629 0.1547 0.1463 0.1375 0.1282 0.1181 0.3 0.2563 0.2453 0.2339 0.2219 0.2092 0.1955 0.1806 0.4 0.3431 0.3296 0.3154 0.3003 0.2839 0.2661 0.2463 0.5 0.4322 0.4168 0.4003 0.3825 0.3628 0.3409 0.3164 0.6 0.5251 0.5085 0.4903 0.4701 0.4475 0.4218 0.3923 0.7 0.6236 0.6066 0.5876 0.5659 0.5408 0.5115 0.4769 0.8 0.7308 0.7149 0.6963 0.6743 0.6478 0.6153 0.5753 0.9 0.8521 0.8400 0.8250 0.8058 0.7807 0.7470 0.7014 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 328 APPENDIX A Table A -2. — 0» as a Function of Hk — (Cont.) AXi = 80° Hk Xim = -90° -85° -80° -75° -70° -65° -60° -55° XiM = -10° -5° 0° 5° 10° 15° 20° 25° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2932 0.2517 0.2198 0.1952 0.1759 0.1604 0.1478 0.1372 0.2 0.4176 0.3809 0.3502 0.3244 0.3024 0.2835 0.2670 0.2524 0.3 0.5154 0.4832 0.4552 0.4307 0.4090 0.3896 0.3720 0.3559 0.4 0.5997 0.5720 0.5472 0.5250 0.5047 0.4860 0.4686 0.4524 0.5 0.6758 0.6525 0.6313 0.6117 0.5935 0.5764 0.5601 0.5446 0.6 0.7465 0.7276 0.7100 0.6936 0.6779 0.6630 0 . 6485 0.6344 0.7 0.8133 0.7988 0.7852 0.7722 0.7596 0.7474 0.7354 0.7234 0.8 0.8774 0.8675 0.8580 0.8488 0.8399 0.8310 0.8220 0.8130 0.9 0.9394 0.9343 0.9294 0.9245 0.9197 0.9148 0.9098 0.9046 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk Xim = -50° -45° -40° -35° -30° -25° -20° -15° XiM = 30° 35° 40° 45° 50° 55° 60° 65° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1281 0.1202 0.1132 0.1068 0.1009 0.0954 0.0902 0.0852 0.2 0.2393 0.2274 0.2164 0.2062 0.1964 0.1870 0.1780 0.1690 0.3 0.3410 0.3270 0.3138 0.3010 0.2887 0.2766 0.2646 0.2526 0.4 0.4369 0.4221 0.4077 0.3936 0.3796 0.3656 0.3515 0.3370 0.5 0.5294 0.5145 0.5000 0.4855 0.4706 0.4554 0.4399 0.4236 0.6 0.6204 0.6064 0.5923 0.5779 0.5631 0.5476 0.5314 0.5140 0.7 0.7113 0.6990 0.6862 0.6730 0.6590 0.6441 0.6280 0.6104 0.8 0.8036 0.7938 0.7836 0.7726 0.7607 0.7476 0.7330 0.7165 0.9 0.8991 0.8932 0.8868 0.8798 0.8719 0.8628 0.8522 0.8396 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk Xim = -10° -5° 0° 5° 10° XiM = 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.0803 0.0755 0.0706 0.0657 0.0606 0.2 0.1601 0.1512 0.1420 0.1325 0.1226 0.3 0.2404 0.2278 0.2148 0.2012 0.1867 0.4 0.3221 0.3064 0.2900 0.2724 0.2535 0.5 0.4065 0.3883 0.3687 0.3475 0.3242 0.6 0.4953 0.4750 0.4528 0.4280 0.4003 0.7 0.5910 0.5693 0.5448 0.5168 0.4846 0.8 0.6976 0.6756 0.6498 0.6191 0.5824 0.9 0.8241 0.8048 0.7802 0.7483 0.7068 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 APPENDIX A Table A -2. — 0» as a Function of Hk — (Cont.) 329 AI, = 90° Hk Xim = -90° XiM = 0° -85° 5° -80° 10° -75° 15° -70° 20° -65° 25° -60° 30° -55° 35° -50° 40° -45° 45° 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0000 0.2871 0.4097 0.5064 0.5903 0.6667 0.7380 0.8060 0.8718 0.9362 1.0000 0.0000 0.2488 0.3754 0.4760 0.5639 0.6441 0.7194 0.7916 0.8618 0.9310 1.0000 0.0000 0.2185 0.3460 0.4490 0.5397 0.6230 0.7018 0.7776 0.8519 0.9257 1.0000 0.0000 0.1944 0.3207 0.4249 0.5175 0.6033 0.6849 0.7641 0.8422 0.9204 1.0000 0.0000 0.1751 0.2988 0.4031 0.4969 0.5846 0.6685 0.7507 0.8324 0.9150 1.0000 0.0000 0.1593 0.2797 0.3833 0.4777 0.5667 0.6527 0.7375 0.8225 0.9094 1.0000 0.0000 0.1462 0.2627 0.3651 0.4596 0.5495 0.6371 0.7242 0.8123 0.9034 1.0000 0.0000 0.1352 0.2475 0.3482 0.4423 0.5328 0.6216 0.7107 0.8018 0.8971 1.0000 0.0000 0.1256 0.2337 0.3324 0.4257 0.5163 0.6061 0.6969 0.7907 0.8903 1.0000 0.0000 0.1172 0.2211 0.3174 0.4097 0.5000 0.5903 0.6826 0.7789 0.8828 1.0000 H, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Xim = -40° XiM = 50° 0.0000 0.1097 0.2093 0.3031 0.3939 0.4837 0.5743 0.6676 0.7663 0.8744 1.0000 35c 55c 0.0000 0.1029 0.1982 0.2893 0.3784 0.4672 0.5577 0.6518 0.7525 0.8648 1.0000 30c 60c 0000 0966 1877 2758 3629 4505 5404 6349 7373 8538 0000 -25c 65c 0000 0906 1775 2625 3473 4333 5223 6167 7203 8407 0000 ■20c 70c 0000 0850 1676 2493 3315 4154 5031 5969 7012 8249 0000 15° 75° 0000 0796 1578 2359 3151 3967 4825 5751 6793 8056 0000 -10c 80c 0000 0743 1481 2224 2982 3770 4603 5510 6540 7815 0000 -5° 85° 0000 0690 1382 2084 2806 3559 4361 5240 6246 7512 0000 0C 90c .0000 .0638 .1282 .1940 .2620 .3333 .4097 .4936 .5903 .7129 .0000 AXi = 100° Hk Xim = -90° -85° -80° -75° -70° -65° -60° -55° -50° XiM = 10° 15° 20° 25° 30° 35° 40° 45° 50° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2804 0.2446 0.2156 0.1920 0.1728 0. 1568 0.1434|0. 1320 0.1221 0.2 0.4007 0.3682 0.3399 0.3151 0.2933 0.2739 0.2565 0.2408 0.2264 0.3 0.4962 0.4671 0.4407 0.4168 0.3948 0.3746 0.3558 0.3382 0.3216 0.4 0.5796 0.5538 0.5300 0.5076 0.4866 0.4667 0.4478 0.4296 0.4119 0.5 0.6559 0.6337 0.6125 0.5924 0.5730 0.5542 0.5359 0.5179 0.5000 0.6 0.7279 0.7093 0.6912 0.6737 0.6565 0.6395 0.6225 0.6054 0.5881 0.7 0.7972 0.7824 0.7679 0.7535 0.7390 0.7245 0.7096 0.6943 0.6784 0.8 0.8650 0.8545 0.8440 0.8333 0.8224 0.8111 0.7994 0.7869 0.7736 0.9 0.9323 0.9266 0.9208 0.9148 0.9086 0.9019 0.8947 0.8868 0.8779 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 330 AXi = 100° APPENDIX A Table A-2. — 0» as a Function of Hk — (Cont.) Hk Xim = -45° -40° -35° -30° -25° -20° -15° -10° XiM = 55° 60° 65° 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1132 0.1053 0.0981 0.0914 0.0852 0.0792 0.0734 0.0677 0.2 0.2131 0.2006 0.1889 0.1776 0.1667 0.1560 0.1455 0.1350 0.3 0 . 3057 0.2904 0.2755 0.2610 0.2465 0.2321 0.2176 0.2028 0.4 0.3946 0.3775 0.3605 0.3435 0.3263 0.3088 0.2907 0.2721 0.5 0.4821 0.4641 0.4458 0.4270 0.4076 0.3875 0.3663 0.3441 0.6 0.5704 0.5522 0 . 5333 0.5134 0.4924 0.4700 0.4462 0.4204 0.7 0.6618 0.6442 0 . 6254 0.6052 0.5832 0.5593 0.5329 0.5038 0.8 0 . 7592 0.7435 0.7261 0.7067|0.6849J0.6601 0.6318 0.5993 0.9 0 . 8680 0.8566 0.8432 0.8272 0.8080 0.7844 0.7554 0.7196 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 AXi = 110° Hk Xim = -90° -85° -80° -75° -70° -65° -60° -55° XiM = 20° 25° 30° 35° 40° 45° 50° 55° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2730 0.2392 0.2113 0.1883 0.1692 0.1531 0.1395 0.1278 0.2 0 . 3907 0.3597 0.3322 0.3077 0.2859 0.2663 0.2486 0.2324 0.3 0.4847 0.4565 0.4305 0.4066 0.3845 0.3638 0.3444 0.3261 0.4 0.5673 0.5420 0.5181 0.4955 0.4740 0.4534 0.4336 0.4143 0.5 0.6436 0.6212 0 . 5998 0.5790 0.5588 0.5390 0.5194 0.5000 0.6 0.7161 0.6970 0.6783 0.6598 0.6415 0.6231 0.6046 0 . 5857 0.7 0.7866 0.7712 0.7557 0.7401 0.7243 0.7081 0.6913 0.6739 0.8 0 . 8566 0.8453 0.8338 0.8219 0.8095 0.7964 0 . 7826 0 . 7676 0.9 0 . 9272 0.9210 0.9144 0.9074 0 . 8998 0.8916 0 . 8825 0.8722 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 . 0000 Hk Xim = -50° -45° -40° -35° -30° -25° -20° XiM = 60° 65° 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1175 0.1084 0.1002 0.0926 0.0856 0.0790 0.0728 0.2 0.2174 0.2036 0.1905 0.1781 0.1662 0.1547 0.1434 0.3 0.3087 0.2919 0.2757 0.2599 0.2443 0.2288 0.2134 0.4 0.3954 0.3769 0.3585 0.3402 0.3217 0.3030 0.2839 0.5 0.4806 0.4610 0.4412 0.4210 0.4002 0.3788 0.3564 0.6 0.5664 0.5466 0.5260 0.5045 0.4819 0.4580 0.4327 0.7 0.6556 0.6362 0.6155 0.5934 0.5695 0.5435 0.5153 0.8 0.7514 0.7337 0.7141 0.6923 0.6678 0.6403 0.6093 0.9 0.8605 0.8469 0.8308 0.8117 0.7887 0.7608 0.7270 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 APPENDIX A Table A -2. — 0; as a Function of Hk — (Cont.) 331 AXi - 120c Hk Xim - -90° -85° -80° -75° -70° -65° -60° XiM = 30° 35° 40° 45° 50° 55° 60° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2649 0.2329 0.2059 0.1833 0.1644 0.1483 0.1346 0.2 0.3798 0.3498 0.3230 0.2988 0.2770 0.2572 0.2391 0.3 0.4719 0.4443 0.4186 0.3946 0.3722 0.3511 0.3311 0.4 0.5535 0.5283 0.5042 0.4812 0.4590 0.4376 0.4169 0.5 0.6294 0.6067 0.5846 0.5630 0.5418 0.5208 0.5000 0.6 0.7022 0.6824 0.6627 0.6431 0.6234 0.6034 0.5831 0.7 0.7739 0.7574 0.7407 0.7237 0.7061 0.6879 0.6689 0.8 0.8461 0.8337 0.8208 0.8072 0.7929 0.7775 0.7609 0.9 0.9207 0.9135 0.9057 0.8973 0.8879 0.8774 0.8654 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Hk JL im — OO -50° -45° -40° -35° -30° XiM = 65° 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1226 0.1121 0.1027 0.0943 0.0865 0.0793 0.2 0.2225 0.2071 0.1928 0.1792 0.1663 0.1539 0.3 0.3121 0.2939 0.2763 0.2593 0.2426 0.2261 0.4 0.3966 0.3766 0.3569 0.3373 0.3176 0.2978 0.5 0.4792 0.4582 0.4370 0.4154 0.3933 0.3706 0.6 0.5624 0.5410 0.5188 0.4958 0.4717 0.4465 0.7 0.6489 0.6278 0 . 6054 0.5814 0.5557 0.5281 0.8 0.7428 0.7230 0.7012 0.6770 0 . 6502 0.6202 0.9 0.8517 0.8356 0.8167 0.7941 0.7671 0.7351 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 AXi = 130° Hk Xim = -90° -85° -80° -75° -70° -65° XiM = 40° 45° 50° 55° 60° 65° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2562 0.2256 0.1995 0.1774 0.1586 0.1425 0.2 0.3678 0.3388 0.3124 0.2884 0.2666 0.2466 0.3 0.4579 0.4306 0.4049 0.3808 0.3580 0.3365 0.4 0.5381 0.5128 0.4882 0.4647 0.4419 0.4197 0.5 0.6132 0.5899 0.5671 0.5446 0.5222 0.5000 0.6 0.6860 0.6652 0.6444 0.6233 0.6020 0.5803 0.7 0.7586 0.7408 0.7226 0.7037 0.6841 0.6635 0.8 0.8332 0.8192 0.8044 0.7886 0.7717 0.7534 0.9 0.9122 0.9036 0.8940 0.8834 0.8713 0.8575 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 332 APPENDIX A Table A-2.— 0» as a Function of Hk — (Cont.) AXi = 130° Hk Xim ■■ 60° -55° -50° -45° -40° XiM = 70° 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.1287 0.1166 0.1060 0.0964 0.0878 0.2 0.2283 0.2114 0.1956 0.1808 0.1668 0.3 0.3159 0.2963 0.2774 0.2592 0.2414 0.4 0.3980 0.3767 0.3556 0.3348 0.3140 0.5 0.4778 0.4554 0.4329 0.4101 0.3868 0.6 0.5581 0.5353 0.5118 0.4872 0.4619 0.7 . 0.6420 0.6192 0.5951 0.5694 0.5421 0.8 0.7334 0.7116 0.6876 0.6612 0.6322 0.9 0.8414 0.8226 0.8005 0.7744 0.7438 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 AX» = 140 0 Hk Xim = -90° -85° -80° -75° -70° XiM = 50° 55° 60° 65° 70° 0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.1 0.2470 0.2174 0.1921 0.1704 0.1518 0.2 0.3550 0.3265 0.3005 0.2767 0.2549 0.3 0.4425 0.4154 0.3897 0.3654 0.3423 0.4 0.5209 0.4952 0.4704 0.4462 0.4226 0.5 0.5949 0.5708 0.5471 0.5235 0.5000 0.6 0.6673 0.6452 0.6230 0.6004 0.5774 0.7 0.7405 0.7210 0.7008 0.6797 0.6577 0.8 0.8170 0.8010 0.7838 0.7652 0.7451 0.9 0.9008 0.8901 0.8780 0.8641 0.8482 1.0 1.0000 1.0000 1.0000 1.0000 1.0000 Hh Xim = -65° -60° -55° -50° XiM = 75° 80° 85° 90° 0.0 0.0000 0.0000 0.0000 0.0000 0.1 0.1359 0.1220 0.1099 0.0992 0.2 0.2348 0.2162 0.1990 0.1830 0.3 0.3203 0.2992 0.2790 0.2595 0.4 0.3996 0.3770 0.3548 0.3327 0.5 0.4765 0.4529 0.4292 0.4051 0.6 0.5538 0.5296 0.5048 0.4791 0.7 0.6346 0.6103 0.5846 0.5575 0.8 0.7233 0.6995 0.6735 0.6450 0.9 0.8296 0.8079 0.7826 0.7530 1.0 1.0000 1.0000 1.0000 1.0000 APPENDIX B PROPERTIES OF THE THREE -BAR-LINEAGE NOMOGRAM This appendix includes a mathematical discussion of the contours of the three-bar-linkage nomogram, and a table of curve coordinates for use in the construction of a nomogram suitable for accurate work. The nomogram itself appears as Fig. B-l. B'l. Contours of Constant 6. — In the (p, r?)-plane the contours of con- stant b are given by Eq. (5-44) : r] = cos-1 (cosh p — ie2i,-p). (1) Since the function cos-1 x is multiple valued, rj is a multiple-valued function of p, for any given b. If (p, rj) is a point on the contour of con- stant b, so is (p, ±r] ± 2kir), for any integral value of k. When b > 0, these points all fall on a single continuous contour; when b < 0, the contour consists of a system of isolated closed curves. In any case the complete contour has an infinite set of horizontal axes of symmetry: v = for, A- = 0, ±1, ±2, ■ • • . (2) Other symmetry properties depend upon the sign of b. Contours for 6 < 0 have a vertical axis of symmetry. When b < 0, I - e26 > 0, (3) and one can define a real constant T by T = - i In (1 - e*>), (4a) e-2r = l - e2b. (46) In terms of the parameter T, Eq. (1) becomes 7) = cos"1 [e~r cosh (p + T7)]. (5) Thus 7](p, b) is unchanged by change of sign of p + T; the contour is symmetric to reflection in the vertical line p = -T = *ln (1 - e26). (6) The contours of constant b > 0 have no vertical axis of symmetry ; the above argument does not apply because T as defined by Eq. (4) is no longer real. One can, however, define a real parameter t by the relation e-2t = e2b _ I (7) 333 334 APPENDIX B In terms of the parameter 2, Eq. (1) becomes 7] = cos-1 [e-* sinh (p + t)]. (8) Change in sign of (p + t) changes the sign of the argument on the right ; ri can then be replaced by (2k + 1)t — 77, where k = 0, ±1, + 2, • • • . It follows that the contours of constant b > 0 have an infinite sequence of centers of symmetry at Pk = -* = iln(e»-l), } Vt = (*+'*)*, k = 0, +1, ±2, • • • . J w The limiting contour, 6=0, has no vertical axis or center of symmetry except at infinity. Its equation is v = cos"1 (hep). (10) This curve intersects the axis rj = 0 at p = In 2, and has no points for which p > In 2. It has horizontal asymptotes * = (A: + i)*-, A- = 0 ± 1, ±2, • • • . (11) B«2. Contours of Constant X. — To study the contours of constant X it is necessary to eliminate b from Eq. (1) and Eq. (545) : p = i In (2 cos X + 2 cosh b) + £b. (12) These equations may be rewritten in an interesting and symmetrical form: 2 cos rj = ep + er*(l - e2b), (13) -2 cos X = eb + e~6(l - e2"). (14) Substitution into Eq. (14) of eb, as given by Eq. (13), leads to the relation v ep cos rj - 1 f . C0S X = (l+6»-2e»coBi,)»' (15) An equivalent but simpler relation, cot X = cos ^ ~ e"P; (sin ^ sin x > 0) (16) sin 77 follows from this by trigonometric rearrangement. For the analysis at hand it is convenient to rewrite Eq. (16) as sin X cos rj — sin rj cos X = sin Xe~p, (17) or sin (X - t?) = sin Xe~p, (sin v sin X > 0). (18) As noted, only that portion of this curve is to be considered for which sin 77 has the same sign as sin X. Fig. B-l. — Three-bar-linkage nomogram. 0.1 -0.05 APPENDIX B 335 Let 0 < X0 < 180°. The contour for which X = X0 must lie only in the region for which sin 77, like sin X0, is positive. This contour is then sin (X0 - t)) = sin X0e-p, (sin 77 > 0). (19) On the other hand, the contour X = X0 — 180° must lie only in the region for which sin 77, like sin (Xa — 180°), is negative; it is the contour sin (X0 - 180°- 77) = sin (X„ - 180°)^, sin 77 < 0 (20a) or sin (X0 - v) = sin XQe~^ (sin 77 < 0). (206) These two contours join smoothly at the origin, forming a continuous curve which approaches the horizontal asymptotes 77 = Xo and 77 = X0 - 180° as p —> 00 . Since sin X0 > 0, we may write the complete curve for any X0 as sin (Xo - v) = e~^-^^Xo\ (21) This is the curve cos 77 = e~p, (22) translated upward by AV = Xo - 90°, (23) and to the left by Ap = - In sin X0. (24) Thus, all of the curves defined by Eq. (21) have the same form, what- ever the value of X0. Equation (22) gives directly the form of the curve for Xo = 90°, consisting of the contours X = 90° and X = -90°. It will be noted that Eq. (22) differs from Eq. (10) only by a reflection in a vertical line and a translation parallel to the p-axis. It follows that the two contours X = X0andX = X0 - 180° form (for any 0 < X0 < 180°) a curve of the same form as the contour 6=0 reflected in a vertical line. B«3. Explanation of Table B«l. — Table B-l gives the coordinates in the (p, 77) -plane of the intersection of the contours of constant 6 and the contours of constant X, for X = 0°, 5°, 10°, • • • , 180°, M6 = -0.50, -0.49, • • • , 0.49, 0.50. Reading the coordinate pairs from associated vertical columns, one can plot any contour of constant 6 ; reading them from a single row, one can plot any contour of constant X. 336 APPENDIX B Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram nb = -0.50 -0.49 -0.48 -0.47 -0.46 -0.45 X, V, V, V, V, 1. v. de- MP + 10 de- HP + 10 de- MP + 10 de- MP + 10 de- MP + 10 de- MP + 10 de- grees grees grees grees grees grees grees 0 10.1193 0.00 10.1218 0.00 10.1242 0.00 10.1267 0.00 10.1293 0.00 10.1319 0.00 5 10.1190 1.20 10.1215 1.22 10.1239 1.24 10.1264 1.26 10.1290 1.29 10.1316 1.31 10 10.1181 2.40 10.1206 2.44 10.1230 2.48 10.1245 2.53 10.1280 2.57 10.1306 2.61 15 10.1167 3.59 10.1191 3.65 10.1215 3.72 10.1239 3.78 10.1264 3.85 10.1290 3.91 20 10.1145 4.77 10.1169 4.85 10.1193 4.94 10.1217 5.02 10.1242 5.11 10.1268 5.20 25 10.1118 5.93 10.1141 6.04 10.1165 6.14 10.1189 6.25 10.1214 6.36 10.1239 6.47 30 10.1084 7.08 10.1107 7.20 10.1131 7.33 10.1155 7.46 10.1179 7.59 10.1203 7.73 35 10.1045 8.20 10.1067 8.35 10.1091 8.50 10.1114 8.65 10.1138 8.80 10.1161 8.96 40 10.1000 9.29 10.1021 9.46 10.1044 9.64 10.1067 9.81 10.1090 9.99 10.1113 10.17 45 10.0948 10.36 10.0969 10.55 10.0991 10.74 10.1013 10.94 10.1035 11.14 10.1058 11.34 50 10.0890 11.38 10.0910 11.60 10.0931 11.81 10.0953 12.03 10.0974 12.25 10.0996 12.48 55 10.0826 12.37 10.0845 12.60 10.0865 12.84 10.0886 13.08 10.0907 13.33 10.0928 13.58 60 10.0756 13.30 10.0774 13.56 10.0793 13.82 10.0812 14.09 10.0833 14.35 10.0852 14.63 65 10.0680 14.19 10.0697 14.47 10.0714 14.75 10.0732 15.04 10.0752 15.33 10.0770 15.62 70 10.0597 15.01 10.0613 15.31 10.0629 15.62 10.0646 15.93 10.0664 16.24 10.0681 16.56 75 10.0508 15.77 10.0523 16.09 10.0538 16.42 10.0553 16.75 10.0569 17.08 10.0585 17.43 80 10.0413 16.45 10.0427 16.79 10.0440 17.14 10.0454 17.49 10.0468 17.85 10.0483 18.22 85 10.0313 17.04 10.0324 17.41 10.0336 17.78 10.0348 18.15 10.0360 18.54 10.0373 18.93 90 10.0207 17.55 10.0216 17.93 10.0226 18.32 10.0236 18.72 10.0246 19.12 10.0257 19.54 95 10.0095 17.95 10.0102 18.35 10.0110 18.76 10.0118 19.18 10.0126 19.60 10.0134 20.04 100 9.9978 18.24 9.9983 18.66 9.9988 19.09 9.9994 19.52 10.0000 19.97 10.0005 20.42 105 9.9857 18.40 9 . 9859 18.84 9.9861 19.29 9.9864 19.74 9 . 9868 20.20 9.9870 20.68 110 9.9732 18.43 9.9731 18.88 9.9730 19.34 9.9730 19.81 9.9730 20.29 9 . 9730 20.78 115 9.9603 18.30 9 . 9599 18.76 9.9595 19.24 9.9591 19.72 9.9588 20.21 9.9586 20.72 120 9.9471 18.02 9.9463 18.49 9.9456 18.97 9.9449 19.46 9.9442 19.96 9.9435 20.48 125 9 . 9338 17.56 9.9327 18.03 9.9316 18.51 9.9305 19.01 9.9294 19.52 9.9283 20.05 130 9.9206 16.91 9.9191 17.38 9.9176 17.86 9.9160 18.36 9.9145 18.87 9.9130 19.40 135 9 . 9074 16.07 9.9055 16.53 9.9036 17.00 9.9016 17.49 9.8997 17.99 9 . 8976 18.52 140 9 . 8946 15.02 9.8923 15.46 9 . 8899 15.92 9.8875 16.39 9.8851 16.88 9 . 8826 17.39 145 9 . 8824 13.75 9.8797 14.17 9 . 8768 14.61 9 . 8740 15.06 9.8711 15.52 9 . 8680 16.01 150 9.8711 12.28 9.8679 12.67 9.8646 13.07 9.8613 13.48 9.8579 13.92 9 . 8543 14.37 155 9.8609 10.61 9 . 8572 10.95 9.8535 11.31 9 . 8498 11.68 9 . 8459 12.06 9.8418 12.46 160 9.8520 8.75 9.8480 9.04 9.8438 9.34 9 . 8397 9.65 9 . 8354 9.98 9 . 8309 10.32 165 9.8447 6.72 9 . 8404 6.95 9.8360 7.18 9.8315 7.43 9.8269 7.68 9.8219 7.95 170 9 . 8394 4.56 9.8348 4.71 9 . 8302 4.88 9.8254 5.05 9 . 8206 5.22 9.8152 5.41 175 9 . 8360 2.30 9.8314 2.38 9 . 8266 2.47 9.8218 2.55 9.8166 2.64 9.8110 2.74 180 9.8349 0.00 9 . 8302 0.00 9.8253 0.00 9.8203 0.00 9.8151 0.00 9.8097 0.00 APPENDIX B 337 Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) u& = -0.44 -0.43 -0.42 -0.41 -0.40 -0.39 X, 1i V, V, V, v< *l, de- MP + 10 de- MP + 10 de- tip + 10 de- MP + 10 de- MP + 10 de- MP + 10 de- grees grees grees grees grees grees grees 0 10.1345 0.00 10.1372 0.00 10.1399 0.00 10.1427 0.00 10.1455 0.00 10.1484 0.00 5 10.1342 1.33 10.1369 1.35 10.1396 1.38 10.1424 1.40 10.1452 1.42 10.1481 1.45 10 10.1332 2.66 10.1359 2.70 10.1386 2.75 10.1414 2.80 10.1442 2.84 10.1471 2.89 15 10.1316 3.98 10.1343 4.05 10.1370 4.12 10.1397 4.19 10.1425 4.26 10.1454 4.33 20 10.1294 5.29 10.1320 5.38 10.1347 5.47 10.1374 5.57 10.1401 5.66 10.1430 5.75 25 10.1264 6.59 10.1290 6.70 10.1317 6.81 10.1344 6.93 10.1371 7.05 10.1399 7.17 30 10.1228 7.86 10.1254 8.00 10.1280 8.14 10.1307 8.28 10.1334 8.42 10.1361 8.56 35 10.1186 9.12 10.1211 9.28 10.1237 9.44 10.1263 9.60 10.1290 9.77 10.1316 9.94 40 10.1137 10.35 10.1162 10.53 10.1187 10.72 10.1212 10.90 10.1238 11.09 10.1264 11.29 45 10.1081 11.55 10.1105 11.75 10.1130 11.96 10.1154 12.17 10.1180 12.39 10.1205 12.61 50 10.1019 12.71 10.1042 12.94 10.1066 13.17 10.1090 13.41 10.1115 13.65 10.1139 13.89 55 10.0949 13.83 10.0972 14.08 10.0994 14.34 10.1018 14.60 10.1042 14.87 10.1065 15.14 60 10.0873 14.90 10.0894 15.18 10.0916 15.46 10.0939 15.75 10.0961 16.04 10.0984 16.34 65 10.0791 15.92 10.0810 16.23 10.0831 16.53 10.0852 16.85 10.0873 17.16 10.0895 17.48 70 10.0701 16.88 10.0719 17.21 10.0738 17.54 10.0758 17.88 10.0778 18.22 10.0800 18.57 75 10.0603 17.77 10.0620 18.13 10.0638 18.53 10.0656 18.85 10.0675 19.22 10.0695 19.59 80 10.0499 18.59 10.0514 18.97 10.0531 19.35 10.0547 19.74 10.0564 20.14 10.0583 20.54 85 10.0387 19.32 10.0401 19.72 10.0416 20.13 10.0430 20.55 10.0446 20.97 10.0463 21.40 90 10.0269 19.95 10.0281 20.38 10.0293 20.82 10.0306 21.26 10.0320 21.71 10.0334 22.16 95 10.0144 20.48 10.0154 20.93 10.0162 21.39 10.0174 21.86 10.0185 22.34 10.0197 22.82 100 10.0012 20.89 10.0020 21.36 10.0027 21.84 10.0035 22.34 10.0043 22.84 10.0053 23.35 105 9.9874 21.16 9.9879 21.66 9.9884 22.16 9.9888 22.68 9.9894 23.21 9 . 9900 23.74 110 9.9731 21.28 9.9732 21.80 9.9734 22.33 9.9735 22.87 9.9738 23.42 9.9740 23.98 115 9.9582 21.24 9.9580 21.77 9.9578 22.32 9.9576 22.88 9.9575 23.45 9.9573 24.04 120 9.9429 21.02 9.9423 21 . 56 9.9417 22.12 9.9411 22.70 9 . 9405 23.29 9 . 9400 23.90 125 9.9273 20.59 9 . 9263 21.15 9.9252 21.72 9.9241 22.31 9.9231 22.91 9.9222 23.53 130 9.9115 19.94 9.9100 20.50 9.9084 21.08 9.9069 21.67 9.9054 22.29 9.9039 22.92 135 9 . 8956 19.06 9.8936 19.61 9.8915 20.19 9.8895 20.78 9.8875 21.39 9 . 8854 22.03 140 9.8801 17.92 9.8775 18.46 9.8749 19.02 9.8723 19.61 9.8697 20.21 9.8670 20.84 145 9.8650 16.51 9.8619 17.03 9.8588 17.57 9.8555 18.14 9.8522 18.72 9 . 8489 19.32 150 9.8508 14.83 9.8471 15.32 9.8435 15.82 9.8395 16.35 9 . 8356 16.90 9.8316 17.47 155 9.8378 12.88 9.8335 13.32 9.8293 13.77 9.8247 14.25 9 . 8202 14.75 9.8155 15.27 160 9.8263 10.67 9.8215 11.05 9.8165 11.44 9.8117 11.85 9 . 8066 12.27 9.8012 12.72 165 9.8169 8.24 9.8118 8.53 9 . 8064 8.84 9 . 8009 9.16 9.7953 9.50 9 . 7894 9.86 170 9.8100 5.60 9 . 8045 5.81 9 . 7990 6.02 9 . 7928 6.25 9.7867 6.49 9.7802 6.74 175 9.8056 2.84 9.8000 2.94 9 . 7943 3.05 9.7877 3.17 9.7814 3.29 9.7747 3.42 180 9.8041 0.00 9.7983 0.00 9.7923 0.00 9.7860 0.00 9.7795 0.00 9.7728 0.00 338 APPENDIX B Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) tib = -0.38 -0.37 -0.36 -0.35 -0.34 -0.33 X, V, V, v< v* v. v. de- MP + 10 de- **P + 10 de- HP + 10 de- MP + 10 de- MP + 10 de- HP + 10 de- grees grees grees grees grees grees grees 0 10.1513 0.00 10.1543 0.00 10.1573 0.00 10.1604 0.00 10.1635 0.00 10.1667 0.00 5 10.1509 1.47 10.1540 1.49 10.1570 1.52 10.1601 1.54 10.1631 1.57 10.1662 1.59 10 10.1499 2.94 10.1529 2.99 10.1559 3.03 10.1590 3.08 10.1621 3.13 10.1652 3.18 15 10.1482 4.40 10.1511 4.47 10.1541 4.54 10.1572 4.62 10.1603 4.69 10.1634 4.77 20 10.1458 5.85 10.1486 5.95 10.1517 6.04 10.1547 6.14 10.1578 6.24 10.1609 6.34 25 10.1427 7.29 10.1455 7.41 10.1485 7.53 10.1515 7.65 10.1546 7.78 10.1577 7.90 30 10.1389 8.71 IX). 1417 8.85 10.1446 9.00 10.1476 9.15 10.1506 9.30 10.1537 9.45 35 10.1344 10.11 10.1371 10.28 10.1400 10.45 10.1429 10.62 10.1459 10.80 10.1489 10.98 40 10.1292 11.48 10.1318 11.68 10.1347 11.87 10.1375 12.07 10.1405 12.28 10.1434 12.48 45 10.1232 12.83 10.1258 13.05 10.1286 13.27 10.1314 13.50 10.1343 13.73 10.1371 13.96 50 10.1165 14.14 10.1191 14.38 10.1217 14.63 10.1245 14.89 10.1273 15.14 10.1301 15.40 55 10.1090 15.41 10.1116 15.68 10.1141 15.96 10.1168 16.24 10.1194 16.52 10.1222 16.81 60 10.1008 16.63 10.1032 16.93 10.1057 17.24 10.1082 17.55 10.1108 17.86 10.1135 18.17 65 10.0918 17.81 10.0941 18.14 10.0965 18.47 10.0989 18.81 10.1014 19.15 10.1040 19.49 70 10.0820 18.92 10.0842 19.28 10.0865 19.64 10.0888 20.01 10.0912 20.38 10.0936 20.75 75 10.0714 19.97 10.0734 20.36 10.0756 20.75 10.0778 21.14 10.0800 21.54 10.0823 21.95 80 10.0600 20.95 10.0620 21.36 10.0639 21.78 10.0659 22.21 10.0680 22.64 10.0701 23.08 85 10.0478 21.84 10.0495 22.28 10.0513 22.73 10.0531 23.19 10.0550 23.65 10.0570 24.12 90 10.0348 22.63 10.0363 23.10 10.0379 23.58 10.0395 24.07 10.0412 24.56 10.0429 25.07 95 10.0209 23.31 10.0222 23.82 10.0235 24.33 10.0249 24.84 10.0264 25.37 10.0279 25.91 100 10.0062 23.87 10.0072 24.40 10 . 0083 24.94 10.0094 25.50 10.0106 26.06 10.0119 26.63 105 9 . 9907 24.29 9.9914 24.85 9.9922 25.42 9.9930 26.01 9.9939 26.60 9.9949 27.20 110 9 . 9744 24.55 9 9748 25.14 9.9752 25.74 9.9757 26.35 9.9763 26.98 9 . 9769 27.62 115 9.9573 24.64 9.9573 25.25 9.9574 25.88 9.9575 26.52 9.9576 27.18 9.9578 27.85 120 9.9395 24.52 9.9391 25.15 9.9387 25.81 9.9384 26.48 9 . 9380 27.16 9.9378 27.87 125 9.9212 24.17 9.9202 24.83 9.9194 25.50 9.9186 26.19 9.9177 26.91 9.9170 27.64 130 9 . 9024 23.57 9.9009 24.24 9 . 8995 24.93 9.8981 25.64 9.8967 26.37 9.8953 27.13 135 9 . 8834 22.68 9.8813 23.36 9.8792 24.06 9.8771 24.78 9.8751 25.53 9.8730 26.30 140 9 . 8642 21.49 9.8615 22.16 9 . 8588 22.86 9 . 8560 23.58 9.8532 24.33 9 . 8505 25.10 145 9 . 8454 19.96 9.8421 20.61 9.8386 21.29 9.8350 22.00 9.8314 22.74 9.8279 23.50 150 9 . 8274 18.07 9 . 8233 18.69 9.8190 19.34 9.8147 20.01 9.8102 20.72 9 . 8057 21.46 155 9.8107 15.81 9.8057 16.38 9.8007 16.97 9.7955 17.60 9 . 7902 18.25 9.7847 18.94 160 9.7958 13.19 9.7899 13.69 9.7842 14.20 9.7781 14.75 9.7719 15.33 9.7655 15.93 165 9.7831 10.24 9.7768 10.64 9.7703 11.05 9.7637 11.49 9.7566 11.96 9.7493 12.45 170 9.7737 7.00 9 . 7669 7.28 9.7598 7.57 9.7526 7.88 9.7450 8.21 9.7370 8.56 175 9.7678 3.56 9.7606 3.70 9.7532 3.85 9.7455 4.01 9.7375 4.18 9.7285 4.37 180 9.7658 0.00 9.7585 0.00 9 . 7509 0.00 9.7430 0.00 9.7347 0.00 9.7261 0.00 APPENDIX B 339 Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) nb = -0.32 -0.31 -0.30 -0.29 -0.28 -0.27 X, v. V, V, V< V, V, de- MP + 10 de- HP + 10 de- HP + 10 de- MP + 10 de- MP+ 10 de- MP + 10 de- grees grees grees grees grees grees grees 0 10.1699 0.00 10.1731 0.00 10.1764 0.00 10.1798 0.00 10.1832 0.00 10.1867 0.00 5 10.1695 1.62 10.1728 1.64 10.1760 1.67 10.1795 1.69 10.1829 1.72 10.1863 1.75 10 10.1684 3.23 10.1716 3.28 10.1749 3.34 10.1784 3.39 10.1818 3.44 10.1852 3.49 15 10.1666 4.84 10.1698 4.92 10.1731 5.00 10.1764 5.07 10.1799 5.15 10.1833 5.23 20 10.1641 6.44 10.1673 6.54 10.1705 6.65 10.1738 6.75 10.1772 6.85 10.1806 6.96 25 10.1608 8.03 10.1640 8.16 10.1672 8.29 10.1705 8.42 10.1738 8.55 10.1772 8.68 30 10.1568 9.60 10.1599 9.76 10.1632 9.91 10.1664 10.07 10.1697 10.23 10.1730 10.39 35 10.1520 11.16 10.1551 11.34 10.1582 11.52 10.1615 11.70 10.1647 11.89 10.1680 12.07 40 10.1464 12.69 10.1495 12.89 10.1526 13.10 10.1557 13.32 10.1590 13.53 10.1623 13.74 45 10.1401 14.19 10.1431 14.43 10.1462 14.66 10.1493 14.90 10.1525 15.14 10.1557 15.39 50 10.1330 15.66 10.1359 15.93 10.1389 16.19 10.1420 16.46 10.1451 16.73 10.1482 17.01 55 10.1250 17.10 10.1279 17.39 10.1308 17.69 10.1338 17.98 10.1368 18.28 10.1399 18.59 60 10.1162 18.49 10.1190 18.82 10.1218 19.14 10.1247 19.47 10.1277 19.80 10.1307 20.13 65 10.1066 19.84 10.1093 20.19 10.1120 20.55 10.1148 20.91 10.1177 21.27 10.1206 21.64 70 10.0961 21.13 10.0987 21.51 10.1013 21.90 10.1039 22.29 10.1067 22.69 10.1095 23.09 75 10.0847 22.36 10.0871 22.78 10.0896 23.20 10.0921 23.62 10.0947 24.05 10.0975 24.49 80 10.0723 23.52 10.0746 23.97 10.0769 24.42 10.0794 24.88 10.0818 25.35 10.0844 25.82 85 10.0590 24.59 10.0611 25.08 10.0633 25.57 10.0656 26.06 10.0678 26.56 10.0703 27.07 90 10.0447 25.58 10.0468 26.09 10.0487 26.62 10.0507 27.15 10.0528 27.69 10.0550 28.24 95 10.0295 26.45 10.0312 27.01 10.0329 27.57 10.0347 28.14 10.0366 28.72 10.0386 29.30 100 10.0132 27.21 10.0147 27.80 10.0161 28.40 10.0177 29.01 10.0193 29.62 10.0211 30.25 105 9.9959 27.82 9.9970 28.45 9.9982 29.09 9.9995 29.74 10.0008 30.40 10.0023 31.07 110 9.9775 28.27 9.9783 28.94 9.9792 29.61 9.9801 30.30 9.9811 31.01 9 . 9822 31.72 115 9.9581 28.54 9.9585 29.24 9.9589 29.95 9.9594 30.69 9.9600 31.43 9.9607 32.20 120 9.9376 28.59 9.9376 29.32 9.9375 30.08 9.9376 30.85 9.9377 31.64 9.9379 32.45 125 9.9163 28.39 9.9157 29.16 9.9151 29.95 9.9146 30.76 9.9142 31.59 9.9138 32.45 130 9.8941 27.90 9.8929 28.70 9.8916 29.53 9.8905 30.37 9.8894 31.24 9 . 8884 32.14 135 9.8711 27.09 9.8691 27.91 9.8672 28.76 9.8653 29.64 9.8635 30.54 9.8617 31.48 140 9.8477 25.91 9.8449 26.74 9.8421 27.61 9.8394 28.50 9.8367 29.43 9.8341 30.39 145 9.8242 24.30 9.8205 25.13 9.8168 26.00 9.8131 26.90 9.8094 27.83 9 . 8057 28.81 150 9.8011 22.23 9.7964 23.04 9.7917 23.88 9.7869 24.77 9.7820 25.69 9.7771 26.66 155 9.7791 19.66 9.7734 20.42 9.7675 21.21 9.7615 22.05 9.7553 22.93 9 . 7492 23.85 160 9.7589 16.57 9.7522 17.24 9.7452 17.95 9.7379 18.71 9.7305 19.50 9.7230 20.34 165 9.7418 12.97 9.7339 13.53 9.7260 14.11 9.7175 14.74 9.7088 15.40 9 . 6998 16.11 170 9.7286 8.93 9.7198 9.33 9.7109 9.75 9.7015 10.20 9.6917 10.68 9.6817 11.19 175 9.7200 4.56 9.7113 4.76 9.7010 4.99 9.6913 5.22 9.6804 5.48 9 . 6696 5.75 180 9.7172 0.00 9 . 7078 0.00 9 . 6979 0.00 9 . 6877 0.00 9.6769 0.00 9 . 6656 0.00 340 APPENDIX B Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) nb = -0.26 -0.25, -0.24 -0.23 -0.22 -0.21 X, V, V, V, V, n. V< de- HP + 10 de- fip + 10 de- MP + 10 de- HP + 10 de- MP+ 10 de- HP + 10 de- grees grees grees grees grees grees grees 0 10.1902 0.00 10.1938 0.00 10.1974 0.00 10.2011 0.00 10.2048 0.00 10.2086 0.00 5 10.1898 1.77 10.1935 1.80 10.1970 1.83 10.2006 1.85 10 . 2045 1.88 10.2082 1.91 10 10.1886 3.54 10.1923 3.60 10.1959 3.65 10.1995 3.70 10.2033 3.76 10.2071 3.81 15 10.1868 5.31 10.1903 5.39 10.1940 5.47 10.1976 5.55 10.2013 5.63 10.2051 5.71 20 10.1841 7.07 10.1876 7.17 10.1912 7.28 10.1949 7.39 10.1985 7.50 10 . 2023 7.61 25 10.1807 8.81 10.1842 8.95 10.1878 9.08 10.1914 9.22 10.1951 9.35 10.1988 9.49 30 10.1765 10.55 10.1800 10.71 10.1835 10.87 10.1871 11.03 10.1908 11.20 10.1944 11.36 35 10.1715 12.26 10.1749 12.45 10.1784 12.64 10.1819 12.84 10.1856 13.03 10.1892 13.22 40 10.1656 13.96 10.1690 14.18 10.1725 14.40 10.1760 14.62 10.1796 14.84 10.1832 15.07 45 10.1590 15.63 10.1623 15.88 10.1657 16.13 10.1692 16.38 10.1728 16.63 10.1763 16.89 50 10.1515 17.28 10.1547 17.56 10.1581 17.84 10.1615 18.12 10.1650 18.40 10.1686 18.69 55 10.1430 18.89 10.1463 19.20 10.1496 19.51 10.1529 19.83 10.1563 20.14 10.1598 20.46 60 10.1338 20.47 10.1370 20.81 10.1401 21.16 10.1434 21.50 10.1467 21.85 10.1502 22.20 65 10.1236 22.01 10.1267 22.38 10.1298 22.76 10.1329 23.14 10.1362 23.52 10.1396 23.91 70 10.1124 23.49 10.1154 23.90 10.1184 24.31 10.1215 24.73 10.1246 25.15 10.1279 25.57 75 10.1002 24.93 10.1031 25.37 10.1060 25.82 10.1091 26.26 10.1120 26.73 10.1151 27.19 80 10.0870 26.29 10.0897 26.77 10.0925 27.26 10.0954 27.75 10.0983 28.24 10.1013 28.74 85 10.0727 27.58 10.0753 28.10 10.0779 28.63 10.0806 29.16 10.0834 29.70 10.0862 30.24 90 10.0573 28.79 10.0597 29.35 10.0621 29.92 10.0646 30.49 10.0673 31.07 10.0700 31.66 95 10.0407 29.90 10.0428 30.50 10.0451 31.11 10.0474 31.73 10.0498 32.36 10.0524 32.99 100 10.0229 30.89 10.0248 31.54 10.0268 32.19 10.0289 32.86 10.0311 33.53 10.0334 34.22 105 10.0037 31.75 10.0054 32.44 10.0071 33.15 10.0089 33.86 10.0108 34.59 10.0128 35.31 110 9 . 9833 32.45 9.9846 33.20 9.9860 33.95 9.9875 34.72 9.9891 35.50 9 . 9908 36.29 115 9.9615 32.97 9.9624 33.76 9.9634 34.57 9.9645 35.39 9.9657 36.23 9.9670 37.08 120 9.9383 33.27 9.9386 34.12 9.9392 34.98 9.9398 35.86 9 . 9406 36.75 9.9414 37.67 125 9.9136 33.32 9.9135 34.21 9.9134 35.13 9.9135 36.07 9.9137 37.03 9.9140 38.01 130 9.8875 33.06 9.8867 34.00 9.8859 34.98 9 . 8854 35.97 9.8849 36.99 9.8846 38.04 135 9.8600 32.44 9.8584 33.43 9 . 8569 34.45 9.8555 35.51 9.8542 36.59 9.8530 37.71 140 9.8314 31.39 9.8288 32.42 9.8264 33.48 9.8240 34.59 9.8217 35.73 9.8195 36.91 145 9.8020 29.82 9.7983 30.88 9.7946 31.98 9.7911 33.12 9.7876 34.31 9.7841 35.55 150 9.7722 27.67 9.7671 28.73 9.7622 29.84 9.7571 31.00 9.7523 32.21 9.7472 33.49 155 9.7428 24.83 9.7364 25.86 9.7298 26.94 9.7231 28.09 9.7165 29.29 9.7096 30.57 160 9.7150 21.24 9 . 7069 22.19 9 . 6986 23.20 9.6901 24.27 9.6815 25.41 9.6725 26.63 165 9.6906 16.86 9 . 6808 17.67 9 . 6707 18.54 9.6603 19.46 9.6495 20.46 9 . 6383 21.53 170 9.6708 11.75 9 . 6599 12.34 9 . 6480 12.99 9.6359 13.68 9.6231 14.43 9 . 6097 15.25 175 9 . 6582 6.04 9 . 6459 6.36 9.6333 6.70 9.6197 7.08 9.6052 7.49 9 . 5905 7.93 180 9 . 6537 0.00 9.6411 0.00 9 . 6279 0.00 9.6140 0.00 9.5993 0.00 9.5837 0.00 APPENDIX B 341 Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) nb = -0.20 -0.19 -0.18 -0.17 -0.16 -0.15 X, V, v> V, v< V, v< de- MP + 10 de- MP + 10 de- HP + 10 de- ^P + 10 de- ^P + 10 de- iip + 10 de- grees grees grees grees grees grees grees 0 10.2124 0.00 10.2163 0.00 10.2203 0.00 10.2243 0.00 10.2284 0.00 10.2325 0.00 5 10.2121 1.93 10.2160 1.96 10.2199 1.99 10.2238 2.02 10.2280 2.04 10.2320 2.07 10 10.2109 3.87 10.2147 3.92 10.2189 3.98 10.2227 4.03 10.2268 4.09 10 . 2309 4.14 15 10.2089 5.79 10.2128 5.88 10.2167 5.96 10.2207 6.04 10.2247 6.13 10 . 2288 6.21 20 10.2061 7.72 10.2099 7.83 10.2139 7.94 10.2179 8.05 10.2219 8.16 10.2260 8.27 25 10.2026 9.63 10.2065 9.77 10.2103 9.91 10.2143 10.05 10.2183 10.19 10.2224 10.33 30 10.1982 11.53 10.2021 11.70 10.2059 11.87 10.2098 12.04 10.2139 12.21 10.2179 12.38 35 10.1930 13.42 10.1968 13.62 10.2006 13.81 10.2045 14.01 10.2085 14.21 10.2126 14.41 40 10.1869 15.29 10.1907 15.52 10.1945 15.75 10.1984 15.98 10.2023 16.21 10.2064 16.44 45 10.1800 17.15 10.1838 17.41 10.1875 17.66 10.1914 17.92 10.1952 18.19 10.1992 18.45 50 10.1722 18.98 10.1759 19.27 10.1796 19.56 10.1834 19.85 10.1872 20.15 10.1912 20.44 55 10.1634 20.78 10.1670 21.11 10.1707 21.43 10.1744 21.76 10.1783 22.08 10.1821 22.41 60 10.1536 22.56 10.1572 22.91 10.1609 23.27 10.1645 23.63 10.1684 24.00 10.1721 24.36 65 10.1429 24.30 10.1464 24.69 10.1501 25.09 10.1536 25.48 10.1574 25.88 10.1611 26.28 70 10.1312 26.00 10.1346 26.43 10.1381 26.86 10.1416 27.29 10.1453 27.73 10.1490 28.17 75 10.1184 27.65 10.1216 28.12 10.1250 28.59 10.1285 29.07 10.1320 29.54 10.1357 30.02 80 10.1044 29.25 10.1075 29.76 10.1109 30.27 10.1142 30.79 10.1177 31.31 10.1212 31.84 85 10.0892 30.79 10.0922 31,34 10.0954 31.90 10.0986 32.46 10.1020 33.02 10.1054 33.59 90 10.0728 32.25 10.0756 32.85 10.0786 33.45 10.0817 34.06 10.0849 34.68 10.0882 35.30 95 10.0549 33.64 10.0576 34.28 10.0605 34.93 10.0634 35.59 10.0665 36.26 10.0696 36.93 100 10.0358 34.91 10.0382 35.61 10.0409 36.32 10.0436 37.03 10.0465 37.75 10.0494 38.48 105 10.0150 36.07 10.0172 36.82 10.0196 37.59 10.0221 38.36 10.0247 39.15 10.0274 39.94 110 9.9926 37.09 9.9945 37.91 9.9966 38.73 9.9988 39.57 10.0012 40.42 10.0036 41.28 115 9.9684 37.95 9.9700 38.83 9.9718 39.72 9.9736 40.62 9.9757 41.54 9.9778 42.47 120 9.9424 38.60 9.9436 39.55 9 . 9449 40.51 9.9463 41.49 9.9479 42.49 9.9497 43.50 125 9.9144 39.01 9.9151 40.03 9.9158 41.07 9.9168 42.13 9.9178 43.22 9.9191 44.32 130 9.8843 39.12 9.8843 40.21 9.8844 41.34 9.8847 42.49 9.8851 43.67 9 . 8859 44.86 135 9.8520 38.85 9.8512 40.03 9.8505 41.24 9.8499 42.49 9 . 8496 43.76 9 . 8495 45.07 140 9.8175 38.13 9.8156 39.39 9.8139 40.68 9.8123 42.03 9.8110 43.41 9 . 8098 44.84 145 9.7808 36.84 9.7776 38.17 9.7746 39.56 9.7717 40.99 9.7691 42.48 9.7666 44.03 150 9.7424 34.82 9.7375 36.22 9.7329 37.67 9.7283 39.20 9.7239 40.79 9.7197 42.46 155 9.7028 31.92 9 . 6960 33.34 9.6892 34.84 9 . 6824 36.42 9.6758 38.09 9 . 6692 39.86 160 9.6634 27.93 9.6542 29.31 9.6448 30.79 9.6352 32.38 9.6257 34.07 9.6160 35.88 165 9.6267 22.69 9.6147 23.94 9.6022 25.30 9.5896 26.76 9 . 5763 28.36 9 . 5629 30.09 170 9.5957 16.14 9.5810 17.11 9.5656 18.18 9 . 5492 19.36 9.5322 20.66 9.5143 22.10 175 9.5746 8.42 9.5579 8.96 9 . 5400 9.56 9 . 5208 10.23 9.5005 10.98 9 . 4789 11.82 180 9.5671 0.00 9.5494 0.00 9 . 5306 0.00 9.5104 0.00 9.4888 0.00 9.4655 0.00 342 APPENDIX B Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) nb - -0.14 -0.13 • -0.12 -0.11 -0.10 -0.09 X, V> V, V, V, V, V, de- y-V + 10 de- fip + 10 de- MP + 10 de- MP + 10 de- HP + 10 de- MP+ 10 de- grees grees grees grees grees grees grees 0 10.2367 0.00 10.2409 0.00 10.2452 0.00 10.2495 0.00 10.2539 0.00 10.2584 0.00 5 10.2363 2.10 10.2405 2.13 10.2447 2.16 10.2491 2.19 10.2535 2.21 10.2580 2.24 10 10.2350 4.20 10.2393 4.26 10.2435 4.31 10.2479 4.37 10.2523 4.43 10.2567 4.48 15 10.2330 6.30 10.2372 6.38 10.2415 6.47 10.2458 6.55 10.2503 6.64 10.2547 6.72 20 10.2301 8.39 10.2344 8.50 10.2386 8.61 10.2429 8.73 10.2474 8.84 10.2518 8.96 25 10.2265 10.47 10.2308 10.61 10.2350 10.76 10.2393 10.90 10.2437 11.04 10.2480 11.19 30 10.2220 12.55 10.2262 12.72 10.2305 12.89 10.2347 13.07 10.2391 13.24 10.2435 13.42 35 10.2166 14.62 10.2208 14.82 10.2250 15.02 10.2293 15.23 10.2336 15.43 10.2380 15.63 40 10.2104 16.67 10.2145 16.91 10.2187 17.14 10.2230 17.37 10.2273 17.61 10.2316 17.85 45 10.2032 18.71 10.2073 18.98 10.2115 19.25 10.2158 19.51 10.2200 19.78 10.2244 20.05 50 10.1951 20.74 10.1992 21.04 10.2034 21.34 10.2076 21.64 10.2118 21.94 10.2162 22.24 55 10.1861 22.75 10.1901 23.08 10.1942 23.41 10.1984 23.75 10.2026 24.09 10.2070 24.42 60 10.1761 24.73 10.1800 25.10 10.1840 25.47 10.1882 25.84 10.1924 26.21 10.1967 26.59 65 10.1650 26.69 10.1688 27.09 10.1729 27.50 10.1770 27.91 10.1811 28.32 10.1853 28.74 70 10.1528 28.62 10.1565 29.06 10.1605 29.51 10.1646 29.96 10.1687 30.41 10.1728 30.87 75 10.1394 30.51 10.1431 31.00 10.1470 31.49 10.1510 31.98 10.1550 32.47 10.1592 32.97 80 10.1248 32.36 10.1285 32.89 10.1323 33.43 10.1362 33.97 10.1402 34.51 10.1443 35.05 85 10.1089 34.17 10.1125 34.75 10.1162 35.33 10.1200 35.92 10.1239 36.50 10.1279 37.10 90 10.0916 35.92 10.0951 36.55 10.0987 37.18 10.1024 37.81 10.1062 38.46 10.1101 39.11 95 10.0728 37.61 10.0762 38.29 10.0797 38.98 10.0832 39.67 10.0869 40.37 10.0908 41.07 100 10.0524 39.22 10.0557 39.96 10.0590 40.71 10.0624 41.46 10.0660 42.22 10.0697 42.99 105 10.0303 40.74 10.0334 41.54 10.0364 42.36 10.0397 43.18 10.0431 44.00 10.0467 44.84 110 10.0063 42.14 10.0091 43.02 10.0120 43.91 10.0150 44.80 10.0182 45.70 10.0216 46.61 115 9.9801 43.42 9.9826 44.37 9.9853 45.34 9 . 9880 46.32 9.9910 47.30 9 . 9942 48.29 120 9.9516 44.53 9.9538 45.57 9.9561 46.62 9.9586 47.69 9.9612 48.77 9.9642 49.86 125 9 . 9206 45.44 9.9223 46.57 9.9242 47.73 9.9263 48.90 9.9286 50.08 9.9312 51.28 130 9 . 8867 46.09 9.8879 47.33 9.8892 48.60 9.8908 49.88 9 . 8926 51.19 9 . 8947 52.52 135 9.8496 46.41 9.8500 47.77 9.8507 49.16 9.8516 50.59 9 . 8528 52.03 9.8543 53.50 140 9.8090 46.30 9.8084 47.80 9.8081 49.34 9 . 8082 50.91 9.8085 52.52 9.8092 54.16 145 9.7645 45.62 9.7626 47.27 9.7610 48.97 9.7598 50.72 9.7590 52.52 9.7586 54.37 150 9.7157 44.19 9.7121 46.00 9.7088 47.87 9.7059 49.82 9.7034 51.84 9.7014 53.93 155 9.6629 41.72 9.6568 43.67 9.6510 45.73 9.6457 47.89 9 . 6407 50.16 9 . 6364 52.53 160 9.6064 37.82 9.5970 39.89 9.5877 42.10 9 . 5787 44.46 9.5701 46.97 9 . 5620 49.65 165 9.5490 31.99 9.5349 34.05 9 . 5207 36.30 9 . 5064 38.76 9.4921 41.46 9 . 4782 44.39 170 9.4955 23.70 9 . 4758 25.49 9.4552 27.51 9 . 4336 29.78 9.4113 32.35 9.3881 35.28 175 9.4555 12.78 9.4303 13.88 9.4034 15.14 9.3739 16.62 9.3422 18.35 9 . 3076 20.42 180 9 . 4402 0.00 9.4128 0.00 9.3828 0.00 9.3498 0.00 9.3132 0.00 9.2722 0.00 APPENDIX B 343 Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) nb = -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 X, v> V, V, V, V, V, de- MP + 10 de- P-V + 10 de- MP + 10 de- MP+ 10 de- MP + 10 de- PP + 10 de- grees grees grees grees grees grees grees 0 10.2629 0.00 10.2674 0.00 10.2721 0.00 10.2768 0.00 10.2815 0.00 10.2863 0.00 5 10.2625 2.27 10.2670 2.30 10.2717 2.33 10.2764 2.36 10.2811 2.39 10.2860 2.41 10 10.2613 4.54 10.2658 4.60 10.2704 4.66 10.2751 4.71 10.2799 4.77 10.2846 4.83 15 10.2592 6.81 10.2637 6.89 10.2683 7.98 10.2730 7.07 10.2778 7.15 10.2825 7.24 20 10.2563 9.07 10.2608 9.19 10.2654 9.30 10.2701 9.42 10.2748 9.54 10.2797 9.65 25 10.2526 11.33 10.2571 11.48 10.2617 11.62 10.2664 11.77 1,0.2710 11.92 10 . 2760 12.06 30 10.2480 13.59 10.2525 13.77 10.2571 13.94 10.2617 14.12 10.2665 14.29 10.2713 14.47 35 10.2425 15.84 10.2470 16.05 10.2516 16.25 10.2562 16. 4C 10.2610 16.67 10.2657 16.87 40 10.2361 18.09 10.2407 18.32 10.2452 18.56 10.2499 18.80 10.2545 19.04 10.2593 19.28 45 10.2288 20.32 10.2333 20.59 10.2379 20.86 10.2425 21.14 10.2472 21.41 10.2520 21.68 50 10.2205 22.55 10.2250 22.85 10.2296 23.16 10.2342 23. 4C 10.2389 23.77 10.2437 24.08 55 10.2112 24.76 10.2158 25.10 10.2203 25.44 10.2249 25.79 10.2295 26.13 10.2343 26.47 60 10.2010 26.97 10.2055 27.34 10.2099 27.72 10.2146 28.10 10.2191 28.48 10.2239 28.86 65 10.1897 29.15 10.1941 29.57 10.1985 29.99 10.2031 30.40 10.2077 30.82 10.2124 31.24 70 10.1771 31.32 10.1815 31.78 10.1859 32.24 10.1904 32.69 10.1951 33.16 10.1998 33.61 75 10.1634 33.47 10.1677 33.97 10.1721 34.47 10.1766 34.97 10.1812 35.48 10.1859 35.98 80 10. 1484 35.59 10.1527 36.14 10.1570 36.69 10.1615 37.24 10.1661 37.79 10.1708 38.34 85 10.1320 37.69 10.1362 38.29 10.1405 38.89 10.1449 39.49 10.1495 40.08 10.1541 40.69 90 10.1142 39.75 10.1183 40.40 10.1225 41.06 10.1270 41.71 10.1314 42.37 10.1360 43.02 95 10.0947 41.78 10.0988 42.49 10.1029 43.20 10.1073 43.91 10.1117 44.62 10.1163 45.34 100 10.0735 43.76 10.0775 44.53 10.0816 45.30 10.0858 46.08 10.0902 46.86 10.0948 47.64 105 10.0504 45.67 10.0542 46.52 10.0583 47.36 10.0624 48.21 10.0667 49.07 10.0712 49.92 110 10.0252 47.53 10.0289 48.45 10.0328 49.37 10.0368 50.31 10.0410 51.24 10 . 0454 52.18 115 9.9976 49.30 10.0011 50.31 10.0048 51.32 10.0087 52.34 10.0128 53.37 10.0172 54.40 120 9.9672 50.96 9.9706 52.07 9.9741 53.19 9.9778 54.31 9.9818 55.44 9 . 9860 56.58 125 9.9340 52.49 9.9370 53.72 9.9403 54.95 9.9438 56.20 9.9476 57.45 9.9517 58.71 130 9.8971 53.86 9.8998 55.22 9.9028 56.59 9.9060 57.97 9.9096 59.36 9.9135 60.77 135 9.8562 55.00 9.8583 56.51 9.8609 58.05 9.8638 59.60 9 . 8670 61.16 9 . 8706 62.74 140 9.8103 55.84 9.8119 57.54 9.8138 59.27 9.8162 61.02 9.8190 62.79 9 . 8223 64.58 145 9.7587 56.26 9.7593 58.19 9.7604 60.16 9.7620 62.17 9.7642 64.20 9.7670 66.25 150 9.7000 56.08 9 . 6992 58.29 9.6990 60.57 9 . 6996 62.89 9.7008 65.26 9 . 7029 67.66 155 9.6326 55.00 9.6297 57.56 9.6275 60.22 9 . 6263 62.96 9.6261 65.77 9 . 6269 68.65 160 9 . 5546 52.49 9.5481 55.48 9 . 5426 58.64 9.5381 62.02 9.5354 65.37 9.5341 68.92 165 9.4647 47.60 9.4520 51.08 9.4404 54.85 9 . 4304 58.91 9.4222 63.23 9.4164 67.81 170 9.3645 38.61 9.3408 42.41 9.3174 46.75 9.2951 51.68 9.2749 57.25 9.2581 63.46 175 9 . 2697 22.93 9.2285 26.00 9.1834 29.84 9.1349 34.71 9.0834 40.99 9.0315 49.16 180 9.2259 0.00 9.1728 0.00 9.1107 0.00 9.0364 0.00 8.9445 0.00 8.8244 0.00 344 APPENDIX B Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) M6 = -0.02 -0.01 0.00 0.01 0.02 X, degrees np + 10 V, degrees MP + 10 degrees MP + 10 V, degrees HP + 10 degrees MP + 10 9i degrees 0 10.2912 0.00 10.2960 0.00 10.3010 0.00 10.3060 0.00 10.3112 0.00 5 10 . 2908 2.44 10.2956 2.47 10.3006 2.50 10.3056 2.53 10.3108 2.56 10 10.2895 4.88 10.2944 4.94 10 . 2994 5.00 10.3044 5.06 10.3095 5.12 15 10.2874 7.33 10.2923 7.41 10.2973 7.50 10.3023 7.59 10.3074 7.67 20 10.2844 9.77 10.2894 9.88 10.2944 10.00 10.2994 10.12 10 . 3044 10.23 25 10.2808 12.21 10.2857 12.35 10.2906 12.50 10.2957 12.65 10.3008 12.79 30 10.2761 14.65 10.2810 14.82 10.2860 15.00 10.2910 15.18 10.2961 15.35 35 10.2706 17.08 10.2755 17.29 10.2805 17.50 10.2855 17.71 10.2906 17.92 40 10.2641 19.52 10.2691 19.76 10.2740 20.00 10.2791 20.24 10.2841 20.48 45 10.2568 21.95 10.2617 22.23 10.2667 22.50 10.2717 22.77 10.2768 23.05 50 10.2485 24.39 10.2533 24.69 10.2583 25.00 10.2633 25.31 10.2685 25.61 55 10.2391 26.81 10.2440 27.16 10.2490 27.50 10.2540 27.84 10.2591 28.19 60 10.2287 29.24 10.2336 29.62 10 . 2386 30.00 10.2436 30.38 10.2487 30.76 65 10.2172 31.66 10.2221 32.08 10.2271 32.50 10.2321 32.92 10.2372 33.34 70 10.2046 34.08 10.2094 34.54 10.2144 35.00 10.2194 35.46 10.2246 35.92 75 10.1907 36.49 10.1946 36.99 10.2005 37.50 10.2046 38.01 10.2107 38.51 80 10.1755 38.89 10.1803 39.45 10.1853 40.00 10.1903 40.55 10.1955 41.11 85 10.1589 41.29 10.1637 41.90 10.1687 42.50 10.1737 43.10 10.1789 43.71 90 10.1407 43.68 10.1456 44.34 10.1505 45.00 10.1556 45.66 10.1607 46.32 95 10.1209 46.06 10.1258 46.78 10.1307 47.50 10.1358 48.22 10.1409 48.94 100 10.0994 48.43 10.1042 49.21 10.1091 50.00 10.1142 50.79 10.1194 51.57 105 10.0758 50.78 10.0797 51.64 10.0855 52.50 10.0897 53.36 10.0958 54.22 110 10.0500 53.12 10.0547 54.06 10.0596 55.00 10.0647 55.94 10.0700 56.88 115 10.0216 55.43 10.0264 56.47 10.0313 57.50 10.0364 58.54 10.0416 59.57 120 9.9904 57.72 9.9951 58.86 10.0000 60.00 10.0051 61.14 10.0104 62.28 125 9.9560 59.97 9.9606 61.23 9.9654 62.50 9.9706 63.77 9.9760 65.03 130 9.9177 62.17 9.9221 63.59 9.9270 65.00 9.9321 66.41 9.9377 67.83 135 9.8747 64.32 9.8791 65.91 9.8839 67.50 9.8891 69.09 9 . 8947 70.68 140 9.8261 66.38 9.8303 68.19 9.8351 70.00 9.8403 71.81 9.8461 73 . 62 145 9.7705 68.33 9.7745 70.41 9.7792 72.50 9.7845 74.59 9.7905 76.67 150 9.7058 70.09 9.7095 72.54 9.7140 75.00 9.7195 77.46 9.7258 79.91 155 9 . 6289 71.57 9.6320 74.53 9.6364 77.50 9.6420 80.47 9 . 6489 83.43 160 9.5344 72.56 9.5367 76.27 9.5407 80.00 9.5467 83.73 9.5544 87.44 165 9.4134 72.58 9.4134 77.50 9.4167 82.50 9.4234 87.50 9.4334 92.42 170 9 . 2460 70.26 9.2401 77.51 9.2413 85.00 9.2501 92.49 9 . 2660 99.74 175 8.9841 59.70 8.9503 72.73 8.9407 87.50 8.9603 102 . 27 9.0041 115.30 180 8.6532 0.00 8.3572 0.00 — 00 90.00 8.3672 180.00 8.6732 180.00 APPENDIX B 345 Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — {Cord.) yb = 0.03 0.04 0.05 0.06 0.07 0.08 X, V, V, V, V, V, V, de- MP + 10 de- MP + 10 de- HP + 10 de- MP+ 10 de- MP+ 10 de- MP+ 10 de- grees grees grees grees grees grees grees 0 10.3163 0.00 10.3215 0.00 10.3268 0.00 10.3321 0.00 10.3374 0.00 10.3429 0.00 5 10.3160 2.59 10.3211 2.61 10.3264 2.64 10.3317 2.67 10.3370 2.70 10.3425 2.73 10 10.3146 5.17 10.3199 5.23 10.3251 5.29 10.3304 5.34 10.3358 5.40 10.3413 5.46 15 10.3125 7.76 10.3178 7.85 10.3230 7.93 10.3283 8.02 10.3337 8.11 10.3392 8.19 20 10.3097 10.35 10.3148 10.46 10.3201 10.58 10.3254 10.70 10.3308 10.81 10.3363 10.93 25 10.3060 12.94 10.3110 13.08 10.3164 13.23 10.3217 13.38 10.3271 13.52 10.3326 13.67 30 10.3013 15.53 10.3065 15.71 10.3117 15.88 10.3171 16.06 10.3225 16.23 10.3280 16.41 35 10.2957 18.13 10.3010 18.33 10.3062 18.54 10.3116 18.75 10.3170 18.95 10.3225 19.16 40 10.2893 20.72 10.2945 20.96 10.2999 21.20 10.3052 21.44 10.3107 21.68 10.3161 21.91 45 10.2820 23.32 10.2872 23.59 10.2925 23.86 10.2979 24.14 10.3033 24.41 10.3088 24.68 50 10.2737 25.92 10.2789 26.23 10.2842 26.54 10.2896 26.84 10.2950 27.15 10.3005 27.45 55 10.2643 28.53 10.2695 28.87 10.2749 29.21 10.2803 29.56 10.2858 29.90 10.2912 30.24 60 10.2539 31.14 10.2591 31.52 10.2646 31.90 10.2699 32.28 10.2755 32.66 10.2810 33.03 65 10.2424 33.76 10.2477 34.18 10.2531 34.60 10.2585 35.01 10.2641 35.43 10.2697 35.85 70 10.2298 36.39 10.2351 36.84 10.2404 37.31 10.2459 37.76 10.2515 38.22 10.2571 38.68 75 10.2159 39.02 10.2212 39.52 10.2266 40.03 10.2321 40.53 10.2377 41.03 10.2434 41.53 80 10.2008 41.66 10.2061 42.21 10.2115 42.76 10.2170 43.31 10.2227 43.86 10.2284 44.41 85 10.1841 44.31 10.1895 44.92 10.1949 45.51 10.2005 46.11 10.2062 46.71 10.2120 47.31 90 10.1660 46.98 10.1714 47.63 10.1770 48.29 10.1825 48.94 10.1883 49.60 10.1942 50.25 95 10.1463 49.66 10.1517 50.38 10.1573 51.09 10.1629 51.80 10.1688 52.51 10.1747 53.22 100 10.1248 52.36 10.1302 53.14 10.1358 53.92 10.1416 54.70 10.1475 55.47 10.1535 56.24 105 10.1012 55.08 10.1067 55.93 10.1124 56.79 10.1183 57.64 10.1242 58.48 10.1304 59.33 110 10.0754 57.82 10.0810 58.76 10.0868 59.69 10.0928 60.63 10.0989 61.55 10.1052 62.47 115 10.0472 60.60 10.0528 61.63 10.0587 62.66 10.0648 63.68 10.0711 64.69 10.0776 65.70 120 10.0160 63.42 10.0218 64.56 10.0278 65.69 10.0341 66.81 10.0406 67.93 10.0472 69.04 125 9.9817 66.29 9.9876 67.55 9.9938 68.80 10.0003 70.05 10.0070 71.28 10.0140 72.51 130 9.9435 69.23 9.9496 70.64 9.9560 72.03 9.9628 73.41 9.9698 74.78 9.9771 76.14 135 9.9006 72.26 9.9070 73.84 9.9138 75.40 9.9209 76.95 9.9283 78.49 9.9362 80.00 140 9.8523 75.42 9.8590 77.21 9.8662 78.98 9.8738 80.73 9.8819 82.46 9 . 8903 84.16 145 9.7970 78.75 9 . 8042 80.80 9.8120 82.83 9.8204 84.84 9.8293 86.81 9.8387 88.74 150 9.7329 82.34 9.7408 84.74 9.7496 87.11 9.7590 89.43 9.7692 91.71 9.7800 93.92 155 9.6569 86.35 9.6661 89.23 9.6763 92.04 9.6875 94.78 9.6997 97.44 9.7126 100.00 160 9.5641 91.08 9 . 5754 94.63 9.5881 97.98 9.6026 101.36 9.6181 104.52 9.6346 107.51 165 9.4464 97.19 9.4622 101.77 9.4804 106.09 9.5004 110.15 9 . 5220 113.92 9 . 5447 117.40 170 9.2881 106.54 9.3149 112.75 9.3451 118.32 9.3774 123.25 9.4108 127 . 59 9.4445 131.39 175 9.0615 125.84 9.1234 134.01 9 . 1849 140.29 9.2434 145.16 9 . 2985 149.00 9.3497 152.07 180 8.8544 180.00 8.9845 180.00 9 . 0864 180.00 9.1707 180.00 9 . 2428 180.00 9 . 3059 180.00 346 APPENDIX B Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) pb = 0.09 0.10 0.11 0.12 0.13 0.14 X, f. V. *. V, v< V, de- MP + 10 de- HP + 10 de- MP+ 10 de- MP + 10 de- MP + 10 de- MP + 10 de- grees grees grees grees grees grees grees 0 10.3484 0.00 10.3539 0.00 10.3595 0.00 10.3652 0.00 10.3709 0.00 10.3767 0.00 5 10.3480 2.76 10.3535 2.79 10.3591 2.81 10.3647 2.84 10.3705 2.87 10.3763 2.90 10 10.3467 5.52 10.3523 5.57 10.3579 5.63 10.3635 5.69 10.3693 5.74 10.3750 5.80 15 10.3447 8.28 10.3503 8.36 10.3558 8.45 10.3615 8.53 10.3672 8.62 10.3730 8.70 20 10.3418 11.04 10.3474 11.16 10.3529 11.27 10.3586 11.39 10.3644 11.50 10.3701 11.61 25 10.3380 13.81 10.3437 13.96 10.3493 14.10 10.3550 14.24 10.3608 14.39 10.3665 14.53 30 10.3335 16.58 10.3391 16.76 10.3447 16.93 10.3505 17.11 10.3562 17.28 10.3620 17.45 35 10.3280 19.37 10.3336 19.57 10.3393 19.77 10.3450 19.98 10.3508 20.18 10.3566 20.38 40 10.3216 22.15 10.3273 22.39 10.3330 22.63 10.3387 22.86 10.3445 23.09 10.3504 23.33 45 10.3144 24.95 10.3200 25.22 10.3258 25.49 10.3315 25.75 10.3373 26.02 10.3432 26.29 50 10.3062 27.76 10.3118 28.06 10.3176 28.36 10.3234 28.66 10.3292 28.96 10.3351 29.26 55 10.2970 30.58 10.3026 30.91 10.3084 31.25 10.3142 31.59 10.3201 31.92 10.3261 32.25 60 10.2867 33.41 10.3924 33.79 10.2982 34.16 10.3040 34.53 10.3100 34.90 10.3161 35.27 65 10.2753 36.26 10.3811 36.68 10.2870 37.09 10.2929 37.50 10.2988 37.91 10.3050 38.31 70 10.2628 39.13 10.2687 39.59 10.2746 40.04 10.2805 40.49 10.2865 40.94 10.2928 41.38 75 10.2492 42.03 10.2550 42.53 10.2610 43.02 10.2670 43.51 10.2731 44.00 10.2794 44.49 80 10.2343 44.95 10.2402 45.49 10.2462 46.03 10.2523 46.57 10.2585 47.11 10.2648 47.64 85 10.2179 47.90 10.2239 48.50 10.2300 49.08 10.2362 49.67 10.2425 50.25 10.2489 50.83 90 10.2001 50.89 10.2062 51.54 10.2124 52.19 10.2187 52.82 10.2251 53.45 10.2316 54.08 95 10.1808 53.93 10.1869 54.63 10.1932 55.33 10.1997 56.02 10.2062 56.71 10.2128 57.39 100 10.1597 57.01 10.1660 57.78 10.1724 58.54 10.1790 59.29 10.1857 60.04 10.1924 60.78 105 10.1367 60.16 10.1431 61.00 10.1497 61.82 10.1564 62.64 10.1634 63.46 10.1703 64.26 110 10.1116 63.39 10.1182 64.30 10.1250 65.20 10.1320 66.09 10.1391 66.98 10.1463 67.86 115 10.0842 66.71 10.0910 67.70 10.0980 68.68 10.1053 69.66 10.1126 70.63 10.1201 71.58 120 10.0542 70.14 10.0612 71.23 10.0686 72.31 10.0761 73.38 10.0838 74.43 10.0916 75.47 125 10.0212 73.72 10.0286 74.92 10.0363 76.10 10.0442 77.27 10.0523 78.43 10.0606 79.56 130 9.9847 77.48 9.9926 78.81 10.0008 80.12 10.0092 81.40 10.0179 82.67 10.0267 83.91 135 9.9443 81.50 9.9528 82.97 9.9616 84.41 9.9707 85.84 9 . 9800 87.23 9 . 9896 88.59 140 9.8992 85.84 9.9085 87.48 9.9182 89.09 9.9281 90.66 9.9384 92.20 9 . 9490 93.70 145 9.8486 90.63 9.8590 92.48 9.8698 94.28 9.8810 96.03 9.8926 97.73 9.9045 99.38 150 9.7914 96.07 9.8034 98.16 9.8159 100.18 9 . 8288 102.13 9.8421 104.00 9 . 8557 105.81 155 9.7264 102.47 9.7407 104.84 9.7557 107.11 9.7710 109 . 27 9.7868 111.33 9 . 8029 113.28 160 9.6520 110.35 9.6701 113.03 9 . 6887 115.54 9.7077 117.90 9.7270 120.11 9.7464 122.18 165 9.5682 120.61 9.5921 123.54 9.6164 126.24 9.6407 128.70 9.6649 130.95 9.6890 133.01 170 9.4781 134.72 9.5113 137.65 9.5436 140.22 9.5752 142.49 9.6058 144.51 9.6355 146.30 175 9.3976 154.58 9.4422 156.65 9 . 4839 158.38 9.5234 159.86 9 . 5603 161.12 9.5955 162.22 180 9.3622 180.00 9.4132 180.00 9.4598 180.00 9.5028 180.00 9.5428 180.00 9 . 5802 180.00 APPENDIX B 347 Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) fib = 0.15 0.16 0.17 0.18 0.19 0.20 X, V, v. V, V, V, V, de- MP + 10 de- MP + 10 de- txv + 10 de- y-V + 10 de- yp + 10 de- yp + 10 de- grees grees grees grees grees grees grees 0 10.3825 0.00 10.3884 0.00 10.3943 0.00 10.4003 0.00 10.4063 0.00 10.4124 0.00 5 10.3820 2.93 10.3880 2.96 10.3938 2.98 10.3999 3.01 10.4060 3.04 10.4121 3.07 10 10.3809 5.86 10.3868 5.91 10.3927 5.97 10.3989 6.02 10 . 4047 6.08 10.4109 6.13 15 10.3788 8.79 10.3847 8.87 10.3907 8.96 10.3967 9.04 10.4028 9.12 10 . 4089 9.21 20 10.3760 11.73 10.3819 11.84 10.3879 11.95 10.3939 12.06 10.3999 12.17 10.4061 12.28 25 10.3724 14.67 10.3783 14.81 10.3843 14.95 10.3903 15.09 10.3965 15.23 10.4026 15.37 30 10.3679 17.62 10.3739 17.79 10.3798 17.96 10.3859 18.13 10.3921 18.30 10 . 3982 18.47 35 10.3626 20.59 10.3685 20.79 10.3745 20.99 10.3806 21.19 10.3868 21.38 10.3930 21.58 40 10.3564 23.56 10.3623 23.79 10.3684 24.02 10.3745 24.25 10.3807 24.48 10.3869 24.71 45 10.3492 26.55 10.3552 26.81 10.3614 27.08 10.3675 27.34 10.3738 27.59 10.3800 27.85 50 10.3412 29.56 10.3472 29.85 10.3534 30.15 10.3596 30.44 10.3659 30.73 10.3722 31.02 55 10.3321 32.59 10.3383 32.92 10.3444 33.24 10.3507 33.57 10.3570 33.89 10.3634 34.22 60 10.3221 35.64 10.3284 36.00 10.3345 36.37 10.3409 36.73 10.3472 37.09 10.3536 37.44 65 .10.3111 38.72 10.3174 39.12 10.3236 39.52 10.3301 39.91 10.3364 40.31 10.3429 40.70 70 10.2990 41.83 10.3053 42.27 10.3116 42.71 10.3181 43.14 10.3246 43.57 10.3312 44.00 75 10.2857 44.98 10.2920 45.46 10.2985 45.93 10.3050 46.41 10.3116 46.88 10.3184 47.35 80 10.2712 48.16 10.2777 48.69 10.2842 49.21 10.2909 49.73 10.2975 50.24 10.3044 50.75 85 10.2554 51.41 10.2620 51.98 10.2686 52.54 10.2754 53.10 10.2822 53.66 10.2892 54.21 90 10.2382 54.70 10.2449 55.32 10.2517 55.94 10.2586 56.55 10.2656 57.15 10.2728 57.75 95 10.2196 58.07 10.2265 58.74 10.2334 59.41 10.2405 60.07 10.2476 60.72 10 . 2549 61.36 100 10.1994 61.52 10.2065 62.25 10.2136 62.97 10.2209 63.68 10.2282 64.39 10.2358 65.09 105 10.1774 65.06 10.1847 65.85 10.1921 66.64 10.1996 67.41 10.2072 68.18 10.2150 68.93 110 10.1536 68.72 10.1612 69.58 10.1688 70.43 10.1766 71.27 10.1845 72.09 10.1926 72.91 115 10.1278 72.53 10.1357 73.46 10.1436 74.38 10.1518 75.28 10.1600 76.17 10.1684 77.05 120 10.0997 76.50 10.1079 77.51 10.1163 78.51 10.1249 79.49 10.1336 80.45 10.1424 81.40 125 10.0691 80.68 10.0778 81.78 10.0868 82.87 10.0958 83.93 10.0151 84.97 10.1144 85.99 130 10.0359 85.14 10.0451 86.33 10.0547 87.51 10.0644 88.66 10.0743 89.79 10.0843 90.88 135 9.9995 89.93 10.0096 91.24 10.0199 92.51 10.0305 93.76 10.0412 94.97 10.0520 96.15 140 9.9598 95.16 9.9710 96.59 9.9823 97.97 9.9939 99.32 10.0056 100.61 10.0175 101.87 145 9.9166 100.97 9.9291 102.52 9.9417 104.01 9.9546 105.44 9.9676 106.83 9 . 9808 108.16 150 9.8697 107.54 9.8839 109.21 9.8983 110.80 9.9129 112.33 9.9275 113.78 9.9424 115.18 155 9.8192 115.14 9.8358 116.91 9 . 8524 118.58 9 . 8692 120.16 9.8860 121.66 9.9028 123 . 08 160 9.7660 124.12 9.7857 125.93 9.8052 127.62 9 . 8248 129.21 9.8442 130.69 9.8634 132.07 165 9.7129 134.91 9.7363 136.64 9.7596 138.24 9.7822 139.70 9.8047 141.06 9.8267 142.31 170 9.6643 147.90 9.6922 149.34 9.7192 150.64 9.7456 151.82 9.7710 152.89 9.7957 153.86 175 9.6289 163.18 9.6605 164.02 9.6908 164.77 9 . 7200 165.44 9.7479 166.04 9.7746 166.58 180 9.6155 180.00 9.6488 180.00 9.6804 180.00 9.7106 180.00 9.7394 180.00 9.7671 180.00 348 APPENDIX B Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — {Cont.) y.b = 0.21 0.22 0.23 0.24 0.25 0.26 X, V, *. •ji V' V> V, de- HP + 1C de- MP+ 10 de- MP + 10 de- MP + 10 de- HP + 10 de- MP + 1C de- grees grees grees grees grees grees grees 0 10.4186 O.OO 10.4248 0.00 10.4311 0.00 10.4374 0.00 10.4438 0.00 10.4502 0.00 5 10.4182 3.09 10.4245 3.12 10.4306 3.15 10.4370 3.17 10.4435 3.20 10.449S 3.23 10 10.4171 6.19 10.4233 6.24 10.4295 6.30 10.4359 6.35 10.4423 6.40 10.4486 6.46 15 10.4151 9.29 10.4213 9.37 10.4276 9.45 10.4340 9.53 10.4403 9.61 10.4468 9.69 20 10.4123 12.39 10.4185 12.50 10.4249 12.61 10.4312 12.72 10.4376 12.83 10.4441 12.93 25 10.4088 15.51 10.4151 15.65 10.4214 15.78 10.4278 15.92 10.4342 16.05 10.4407 16.19 30 10.4044 18.64 10.4108 18.80 10.4171 18.97 10.4235 19.13 10.4300 19.29 10.4365 19.45 35 10.3992 21.78 10.4056 21.97 10.4119 22.16 10.4184 22.36 10.4249 22.55 10.4315 22.74 40 10.3932 24.93 10.3996 25.16 10.4060 25.38 10.4125 25.60 10.4190 25.82 10.4256 26.04 45 10.3863 28.11 10.3928 28.37 10.3992 28.62 10.4057 28.87 10.4123 29.12 10.4190 29.37 50 10.3786 31.31 10.3850 31.60 10.3915 .31.88 10.3981 32.16 10.4047 32.44 10.4115 32.72 55 10.3698 34.54 10.3763 34.86 10.3829 35.17 10.3896 35.49 10.3963 35.80 10.4030 36.11 60 10.3602 37.80 10.3667 38.15 10.3734 38.50 10.3801 38.84 10.3870 39.19 10.3938 39.53 65 10.3496 41.09 10.3562 41.48 10.3629 41.86 10.3698 42.24 10.3767 42.62 10.3836 42.99 70 10.3379 44.43 10.3446 44.85 10.3515 45.27 10.3584 45.69 10.3654 46.10 10.3724 46.51 75 10.3251 47.81 10.3320 48.27 10.3391 48.74 10.3460 49.18 10.3531 49.63 10.3602 50.07 80 10.3113 51.26 10.3183 51.76 10.3254 52.25 10.3325 52.74 10.3397 53.23 10.3470 53.71 85 10.2962 54.76 10.3034 55.30 10.3106 55.84 10.3179 56.37 10.3253 56.90 10.3327 57.42 90 10.2800 58.34 10.2873 58.93 10.2946 59.51 10.3021 60.08 10.3097 60.65 10.3173 61.21 95 10.2624 62.01 10.2698 62.64 10.2774 63.27 10.2851 63.89 10.2929 64.50 10.3007 65.10 100 10.2434 65.78 10.2511 66.47 10.2589 67.14 10.2668 67.81 10.2748 68.46 10.2829 69.11 105 10.2228 69.69 10.2308 70.41 10.2389 71.14 10.2471 71.85 10.2554 72.56 10.2637 73.25 110 10.2008 73.71 10.2091 74.50 10.2175 75.28 10.2260 76.05 10.2346 76.80 10.2433 77.55 115 10.1770 77.92 10.1857 78.77 10.1945 79.61 10.2034 80.43 10.2124 81.24 10.2215 82.03 120 10.1514 82.33 10.1606 83.25 10.1698 84.14 10.1792 85.02 10.1886 85.88 10.1983 86.73 125 10.1240 86.99 10.1337 87.97 10.1435 88.93 10.1534 89.87 10.1635 90.79 10.1736 91.68 130 10.0946 91.96 10.1049 93.01 10.1154 94.03 10.1259 95.02 10.1367 96.00 10.1475 96.94 135 10.0630 97.29 10.0742 98.41 10.0855 99.49 10.0969 100.55 10.1084 101.57 10.1200 102.56 140 10.0295 103.09 10.0417 104.27 10.0540 105.41 10.0664 106.52 10.0788 107.58 10.0914 108.61 145 9.9941 109.45 10.0076 110.69 10.0211 111.88 10.0346 113.02 10.0483 114.12 10.0620 115.18 150 9.9572 116.51 9.9723 117.79 9.9871 119.00 10.0022 120.16 10.0171 121.27 10.0322 122.33 155 9.9196 124.43 9.9365 125.71 9.9531 126.91 9.9698 128.06 9.9864 129.14 10.0028 130.17 160 9.8825 133.37 9.9015 134.59 9.9201 135.73 9.9386 136.80 9.9569 137.81 9.9750 138.76 165 9 . 8483 143.47 9.8695 144.54 9 . 8903 145.54 9.9107 146.46 9.9308 147.33 9.9506 148.14 170 9.8197 154.75 9.8431 155.57 9.8659 156.32 9.8880 157.01 9.9099 157.66 9.9308 158.25 175 9 . 8005 167.07 9.8252 167.51 9 . 8497 167.92 9.8733 168.30 9.8959 168.64 9.9182 168.96 180 9 . 7937 180.00 9.8193 180.00 9 . 8440 180.00 9.8679 180.00 9.8911 180.00 9.9137 180.00 APPENDIX B 349 Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) ^b = 0.27 0.28 0.29 0.30 0.31 0.32 X, V, V, v> v> V, »7. de- HP + 10 de- HP + 10 de- MP + 10 de- »P + 10 de- HP + 10 de- fip + 10 de- grees grees grees grees grees grees grees 0 10.4567 0.00 10.4632 0.00 10.4698 0.00 10.4764 0.00 10.4831 0.00 10 . 4899 0.00 5 10.4563 3.25 10.4629 3.28 10.4695 3.31 10.4760 3.33 10.4828 3.36 10 . 4895 3.38 10 10.4552 6.51 10.4618 6.56 10.4684 6.61 10.4749 6.66 10.4816 6.72 10.4884 6.77 15 10.4533 9.77 10.4598 9.85 10.4664 9.93 10.4731 10.00 10.4798 10.08 10.4866 10.16 20 10.4506 13.04 10.4572 13.15 10.4638 13.25 10.4705 13.35 10.4773 13.46 10.4841 13.56 25 10.4472 16.32 10.4538 16.45 10.4605 16.58 10.4672 16.71 10.4740 16.84 10.4808 16.97 30 10 4430 19.61 10.4497 19.77 10.4564 19.93 10.4632 20.09 10.4699 20.24 10 . 4768 20.40 35 10.4380 22.93 10.4447 23.11 10.4515 23.30 10.4582 23.48 10.4651 23.66 10.4720 23.84 40 10.4323 26.26 10.4390 26.47 10.4457 26.68 10.4526 26.90 10.4595 27.11 10.4664 27.31 45 10.4257 29.61 10.4325 29.86 10.4393 30.10 10.4462 30.34 10.4531 30.57 10.4601 30.81 50 10.4182 32.99 10.4251 33.27 10.4320 33.54 10.4389 33.81 10.4459 34.07 10.4530 34.34 55 10.4099 36.41 10.4168 36.72 10.4238 37.02 10.4308 37.31 10.4379 37.61 10.4450 37.90 60 10.4007 39.87 10.4077 40.20 10.4147 40.53 10.4218 40.86 10.4290 41.18 10.4362 41.51 65 10.3906 43.36 10.3977 43.73 10.4048 44.09 10.4120 44.45 10.4193 44.81 10.4266 45.16 70 10.3795 46.91 10.3867 47.31 10.3939 47.71 10.4013 48.10 10.4087 48.49 10.4161 48.87 75 10.3675 50.51 10.3747 50.95 10.3821 51.38 10.3896 51.80 10.3971 52.22 10.4047 52.64 80 10.3544 54.18 10.3618 54.65 10.3694 55.12 10.3769 55.58 10.3846 56.03 10.3923 56.48 85 10.3403 57.93 10.3478 58.44 10.3556 58.94 10.3633 59.43 10.3711 59.92 10.3790 60.41 90 10.3250 61.76 10.3328 62.31 10.3407 62.85 10.3487 63.38 10.3568 63.91 10.3647 64.42 95 10.3086 65.70 10.3166 66.28 10.3247 66.86 10.3329 67.43 10.3412 67.99 10.3495 68.55 100 10.2911 69.75 10.2993 70.38 10.3077 70.99 10.3161 71.60 10.3247 72.20 10.3332 72.79 105 10.2723 73.93 10.2808 74.60 10.2895 75.26 10.2982 75.91 10.3070 76.55 10.3159 77.18 110 10.2522 78.28 10.2611 78.99 10.2701 79.70 10.2792 80.39 10.2883 81.06 10.2975 81.73 115 10.2307 82.80 10.2400 83.57 10.2494 84.31 10.2589 85.05 10.2685 85.76 10.2781 86.46 120 10.2079 87.55 10.2177 88.36 10.2276 89.15 10.2375 89.92 10.2476 90.68 10.2576 91.41 125 10.1838 92.55 10.1942 93.41 10.2046 94.24 10.2151 95.05 10.2257 95.84 10.2363 96.61 130 10.1584 97.86 10.1694 98.76 10.1805 99.63 10.1916 100.47 10.2029 101.30 10.2141 102.10 135 10.1317 103.52 10.1435 104.46 10.1553 105.36 10.1672 106.24 10.1791 107.09 10.1911 107.91 140 10.1041 109.61 10.1167 110.57 10.1294 111.50 10.1421 112.39 10.1549 113.26 10.1677 114.09 145 10.0757 116.19 10:0894 117.17 10.1031 118.10 10.1168 119.00 10.1305 119.87 10.1442 120.70 150 10.0471 123.34 10.0620 124.31 10.0769 125.23 10.0917 126.12 10.1064 126.96 10.1211 127.77 155 10.0192 131.15 10.0353 132.07 10.0515 132.95 10.0675 133.79 10.0834 134.58 10.0991 135.34 160 9.9930 139.66 10.0105 140.50 10.0179 141.29 10.0452 142.05 10.0622 142.76 10.0789 143.43 165 9.9698 148.89 9.9888 149.60 10.0075 150.26 10.0260 150.89 10.0439 151.47 10.0618 152.03 170 9.9517 158.81 9.9717 159.32 9.9915 159.80 10.0109 160.25 10.0298 160.67 10.0486 161.07 175 9.9396 169.25 9 . 9604 169 . 52 9.9813 169.78 10.0010 170.01 10.0213 170.24 10.0400 170.44 180 9.9356 180.00 9.9569 180.00 9.9777 180.00 9.9979 180.00 10.0178 180.00 10.0372 180.00 1 350 APPENDIX B Table B-l. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) nb = 0.33 0.34 0.35 0.36 0.37 0.38 X, 1i »7. v, V, v, *• de- HP + 10 de- HP + 10 de- HP + 10 de- HP + 10 de- HP + 10 de- HP + 10 de- grees grees grees grees grees grees grees 0 10.4967 0.00 10.5035 0.00 10.5104 0.00 10.5173 0.00 10.5243 0.00 10.5313 0.00 5 10.4962 3.41 10.5031 3.43 10.5101 3.46 10.5170 3.48 10.5240 3.51 10.5309 3.53 10 10.4952 6.82 10.5021 6.87 10.5090 6.92 10.5159 6.97 10.5229 7.01 10 . 5299 7.06 15 10.4934 10.23 10.5003 10.31 10.5072 10.38 10.5141 10.46 10.5211 10.53 10.5282 10.60 20 10.4909 13.66 10.4978 13.76 10.5047 13.86 10.5117 13.96 10.5186 14.05 10.5258 14.15 25 10.4877 17.10 10.4946 17.22 10.5015 17.35 10.5085 17.47 10.5155 17.59 10.5227 17.71 30 10.4837 20.55 10.4906 20.70 10.4976 20.85 10.5046 21.00 10.5117 21.15 10.5189 21.29 35 10.4789 24.02 10.4859 24.20 10.4929 24.38 10.5000 24.55 10.5071 24.72 10.5144 24.89 40 10.4734 27.52 10.4805 27.72 10.4875 27.93 10.4947 28.13 10.5018 28.32 10.5092 28.52 45 10.4671 31.04 10.4743 31.27 10.4814 31.50 10.4886 31.73 10.4958 31.95 10.5032 32.17 50 10.4601 34.60 10.4673 34.86 10.4745 35.11 10.4817 35.37 10.4891 35.62 10.4965 35.86 55 10.4522 38.19 10.4594 38.48 10.4668 38.76 10.4741 39.04 10.4816 39.32 10.4890 39.59 60 10.4435 41.83 10.4508 42.14 10.4582 42.45 10.4657 42.76 10.4732 43.07 10.4808 43.37 65 10.4340 45.51 10.4414 45.85 10.4489 46.19 10.4565 46.53 10.4641 46.86 10.4718 47.19 70 10.4236 49.25 10.4312 49.62 10.4388 49.99 10.4465 50.36 10.4542 50.72 10.4620 51.08 75 10.4123 53.05 10.4200 53.46 10.4278 53.86 10.4356 54.25 10.4434 54.64 10.4514 55.03 80 10.4001 56.92 10.4080 57.36 10.4159 57.79 10.4239 58.22 10.4320 58.64 10.4400 59.05 85 10.3870 60.88 10.3950 61.35 10.4031 61.81 10.4113 62.27 10.4195 62.72 10.4278 63.16 90 10.3729 64.93 10.3812 65.44 10.3895 65.93 10.3979 66.42 10.4063 66.90 10.4148 67.37 95 10.3579 69.09 10.3664 69.63 10.3749 70.16 10.3835 70.67 10.3922 71.18 10.4009 71.69 100 10.3419 73.37 10.3506 73.94 10.3594 74.50 10.3683 75.06 10.3772 75.60 10.3862 76.13 105 10.3249 77.80 10.3339 78.40 10.3430 78.99 10.3522 79.58 10.3614 80.15 10.3707 80.71 110 10.3069 82.38 10.3163 83.02 10.3257 83.65 10.3352 84.26 10.3448 84.86 10.3544 85.45 115 10.2878 87.15 10.2976 87.82 10.3075 88.48 10.3174 89.12 10.3272 89.75 10.3373 90.36 120 10.2678 92.13 10.2780 92.84 10.2884 93.52 10.2987 94.19 10.3091 94.85 10.3195 95.48 125 10.2470 97.36 10.2577 98.09 10.2686 98.81 10.2794 99.50 10.2902 100.17 10.3012 100.83 130 10.2253 102 . 87 10.2367 103.63 10.2481 104.36 10.2595 105.07 10.2709 105.76 10.2824 106.43 135 10.2030 108.70 10.2151 109.47 10.2271 110.22 10.2392 110.94 10.2513 111.64 10.2634 112.32 140 10.1805 114.90 10.1932 115.67 10.2060 116.42 10.2188 117.14 10.2315 117.84 10.2442 118.51 145 10.1579 121.50 10.1714 122.26 10.1850 123.00 10.1986 123.71 10.2121 124.39 10.2254 125.04 150 10.1357 128.54 10.1502 129.28 10.1647 129.99 10.1790 130.66 10.1933 131.31 10.2074 131.93 155 10.1147 136.06 10.1302 136.75 10.1455 137.40 10.1607 138.03 10.1757 138.62 10.1907 139.19 160 10.0955 144.07 10.1119 144.67 10.1281 145.25 10.1442 145.80 10.1599 146.31 10.1758 146.81 165 10.0793 152.55 10.0966 153.04 10.1137 153.51 10.1303 153.95 10.1468 154.36 10.1631 154.76 170 10.0670 161.44 10.0850 161.79 10.1026 162.12 10.1198 162.43 10.1369 162.72 10.1537 163.00 175 10.0585 170.63 10.0775 170.82 10.0955 170.99 10.1132 171.15 10.1306 171.30 10.1478 171.44 180 10.0561 180.00 10.0747 180.00 10.0930 180.00 10.1109 180.00 10.1285 180.00 10.1458 180.00 APPENDIX B 351 Table B-l. — Coordinates of Points on^the Three-Bar-Linkage Nomogram — (Cont.) ,ib = 0.39 0.40 0.41 0.42 0.43 0.44 A*, de- MP + 10 V, de- MP + 10 de- tip + 10 V, de- MP+ 10 de- MP + 10 de- MP + 10 de- grees grees grees grees grees grees grees 0 10.5384 0.00 10.5455 0.00 10.5527 0.00 10.5599 0.00 10.5672 0.00 10.5745 0.00 5 10 10.5381 10.5371 3.55 7.11 10.5452 10.5442 3.58 7.16 10.5524 10.5514 3.60 7.20 10.5596 10.5586 3.62 7.25 10.5669 10.5659 3.65 7.30 10.5742 10.5732 3.67 7.34 15 10.5354 10.67 10.5425 10.74 10.5497 10.81 10.5570 10.88 10.5643 10.95 10.5716 11.02 20 10.5330 14.25 10.5401 14.34 10.5474 14.43 10.5547 14.53 10.5620 14.62 10.5694 14.71 25 10.5299 17.83 10.5371 17.95 10.5444 18.07 10.5517 18.19 10.5590 18.30 10.5664 18.41 30 10.5261 21.44 10.5334 21.58 10.5407 21.72 10.5480 21.86 10.5554 22.00 10.5628 22.14 35 10.5216 25.06 10.5290 25.23 10.5363 25.40 10.5437 25.56 10.5511 25.72 10.5586 25.88 40 10.5164 28.71 10.5238 28.91 10.5312 29.10 10.5387 29.28 10.5462 29.47 10.5537 29.65 45 10.5105 32.39 10.5180 32.61 10.5254 32.83 10.5330 33.04 10.5405 33.25 10.5481 33.45 50 10.5039 36.11 10.5115 36.35 10.5190 36.59 10.5266 36.83 10.5342 37.06 10.5419 37.29 55 10.4965 39.86 10.5042 40.13 10.5118 40.40 10.5194 40.66 10.5272 40.92 10.5349 41.17 60 10.4884 43.66 10.4961 43.96 10.5039 44.25 10.5116 44.54 10.5194 44.82 10.5273 45.10 65 10.4795 47.52 10.4873 47.84 10.4952 48.15 10.5031 48.47 10.5110 48.77 10.5191 49.08 70 10.4700 51.43 10.4778 51.78 10.4858 52.12 10.4938 52.46 10.5019 52.79 10.5101 53.12 75 10.4595 55.41 10.4675 55.78 10.4756 56.15 10.4838 56.47 10.4920 56.87 10.5003 57.23 80 10.4483 59.46 10.4564 59.86 10.4647 60.26 10.4731 60.65 10.4814 61.03 10.4899 61.41 85 10.4363 63.60 10.4446 64.03 10.4530 64.45 10.4616 64.87 10.4701 65.28 10.4787 65.68 90 10.4234 67.84 10.4320 68.29 10.4406 68.74 10.4493 69.18 10.4581 69.62 10.4669 70.05 95 10.4097 72.18 10.4185 72.66 10.4274 73.14 10.4363 73.61 10.4454 74.07 10.4544 74.52 100 10.3953 76.65 10.4043 77.16 10.4135 77.66 10.4227 78.16 10.4320 78.64 10.4412 79.11 105 10.3800 81.26 10.3894 81.79 10.3988 82.32 10.4084 82.84 10.4179 83.34 10.4274 83.84 110 10.3640 86.02 10.3738 86.58 10.3835 87.13 10.3934 87.67 10.4032 88.20 10.4131 88.72 115 10.3473 90.96 10.3575 91.55 10.3676 92.12 10.3778 92.68 10.3880 93.23 10.3982 93.76 120 10.3300 96.10 10.3405 96.71 10.3511 97.30 10.3617 97.88 10.3723 98.44 10.3829 98.98 125 10.3122 101.47 10.3231 102.09 10.3341 102.69 10.3452 103.28 10.3563 103.85 10.3673 104.41 130 10.2939 107.08 10.3054 107.71 10.3169 108.33 10.3284 108.92 10.3400 109.50 10.3515 110.06 135 10.2754 112.97 10.2875 113.61 10.2995 114.22 10.3115 114.81 10.3236 115.39 10.3356 115.94 140 10.2570 119.16 10.2697 119.79 10.2823 120.39 10.2949 120.98 10.3075 121.54 10.3201 122.08 145 10.2389 125.68 10.2522 126.28 10.2655 126.86 10.2788 127.43 10.2919 127.97 10.3050 128.49 150 10.2216 132.53 10.2356 133.10 10.2495 133.65 10.2635 134.18 10.2771 134.68 10.2908 135.17 155 10.2055 139.73 10.2202 140.25 10.2347 140.75 10.2493 141.23 10.2635 141.68 10.2778 142.12 160 10.1912 147.28 10.2066 147.73 10.2217 148.15 10.2365 148.56 10.2515 148.95 10.2663 149.33 165 10.1794 155.14 10.1953 155.50 10.2109 155.84 10.2264 156.16 10.2418 156.47 10.2569 156.76 170 10.1702 163. 20 10.1867 163.51 10.2028 163.75 10.2190 163.98 10.2345 164.19 10.2500 164.40 175 10.1647 171 . 58 10.1814 171.71 10.1977 171.83 10.2143 171.95 10.2300 172.06 10.2456 172.16 180 10.1628 180.00 10.1795 180.00 10.1960 180.00 10.2123 180.00 10.2283 180.00 10.2441 180.00 352 APPENDIX B Table Bl. — Coordinates of Points on the Three-Bar-Linkage Nomogram — (Cont.) m6 = 0.45 0.46 > 0.47 0.48 0.49 0.50 X, V> V< V, Vf V, V> de- W + 10 de- HP + 10 de- HP + 10 de- MP + 10 de- HP+ 10 de- MP+ io de- grees grees grees grees grees grees grees 0 10.5819 0.00 10.5893 0.00 10.5967 0.00 10.6042 0.00 10.6118 0.00 10.6193 0.00 5 10.5816 3.69 10.5890 3.71 10.5964 3.74 10.6039 3.76 10.6115 3.78 10.6190 3.80 10 10.5806 7.39 10.5880 7.43 10.5955 7.47 10.6030 7.52 10.6106 7.56 10.6181 7.60 15 10.5790 11.09 10.5864 11.15 10.5939 11.22 10.6015 11.28 10.6091 11.35 10.6166 11.41 20 10.5768 14.80 10.5842 14.89 10.5917 14.98 10.5993 15.06 10 . 6069 15.15 10.6145 15.23 25 10.5739 18.53 10.5814 18.64 10.5889 18.75 10.5965 18.86 10.6041 18.96 10.6118 19.07 30 10.5703 22.27 10.5779 22.41 10.5855 22.54 10.5931 22.67 10.6007 22.80 10.6084 22.92 35 10.5661 26.04 10.5738 26.20 10.5814 26.35 10.5891 26.50 10.5967 26.65 10.6045 26.80 40 10.5613 29.83 10.5690 30.01 10.5767 30.19 10.5844 30.36 10.5921 30.54 10.6000 30.71 45 10.5558 33.66 10.5635 33.86 10.5713 34.06 10.5791 34.26 10.5869 34.45 10.5948 34.64 50 10.5496 37.52 10.5574 37.75 10.5653 37.97 10.5731 38.19 10.5810 38.40 10.5890 38.62 55 10.5428 41.42 10.5507 41.67 10.5586 41.92 10.5665 42.16 10.5745 42.40 10.5826 42.63 60 10.5352 45.37 10.5433 45.65 10.5512 45.91 10.5593 46.18 10.5674 46.44 10.5756 46.70 65 10.5270 49.38 10.5352 49.67 10.5432 49.96 10.5514 50.25 10.5597 50.53 10 . 5680 50.81 70 10.5181 53.44 10.5264 53.76 10.5346 54.07 10.5429 54.38 10.5513 54.69 10.5597 54.99 75 10.5085 57.57 10.5169 57.92 10.5253 58.25 10.5338 58.58 10.5423 58.91 10.5508 59.23 80 10.4983 61 78 10.5068 62.15 10.5154 62.51 10.5240 62.86 10.5327 63.21 10.5413 63.55 85 10.4873 66.07 10.4960 66.46 10.5048 66.85 10.5136 67.22 10.5224 67.59 10.5313 67.96 90 10.4757 70.46 10.4846 70.88 10.4936 71.28 10.5026 71.68 10.5116 72.07 10.5207 72.45 95 10.4634 74.96 10.4726 75.40 10.4818 75.82 10.4910 76.24 10.5002 76.65 10.5095 77.05 100 10.4505 79.58 10.4600 80.03 10.4694 80.48 10.4788 80.91 10.4883 81.34 10.4978 81.76 105 10.4370 84.32 10.4468 84.80 10.4564 85.26 10.4661 85.71 10.4759 86.16 10.4857 86.60 110 10.4230 89.22 10.4330 89.71 10.4430 90.19 10.4530 90.66 10.4631 91.12 10.4732 91.57 115 10.4086 94.28 10.4188 94.79 10.4291 95.28 10.4395 95.76 10.4499 96.24 10.4603 96.70 120 10.3935 99.52 10.4042 100.04 10.4149 100.54 10.4256 101.03 10.4363 101.51 10.4471 101.98 125 10.3783 104.95 10.3894 105.48 10.4005 105.99 10.4116 106.49 10.4227 106.97 10.4338 107.44 130 10.3630 110.60 10.3745 111.13 10.3860 111.64 10.3976 112.14 10.4091 112.62 10.4206 113.09 135 10.3476 116.48 10.3597 117.01 10.3716 117.51 10.3836 118.00 10.3955 118.47 10.4074 118.93 140 10.3326 122.61 10.3451 123.12 10.3575 123.61 10.3699 124.08 10.3823 124.54 10.3946 124.98 145 10.3180 128.99 10.3311 129.48 10.3440 129.94 10.3568 130.39 10.3697 130.83 10.3824 131.25 150 10.3043 135.63 10.3179 136.08 10.3313 136.52 10.3446 136.93 10.3579 137.33 10.3711 137.72 155 10.2918 142.54 10.3059 142.94 10.3198 143.32 10.3335 143.69 10.3472 144.05 10.3609 144.39 160 10.2809 149.68 10.2954 150.02 10.3097 150.35 10.3238 150.66 10.3380 150.96 10.3520 151.25 165 10.2719 157.05 10.2869 157.32 10.3015 157.57 10.3160 157.82 10.3304 158.05 10.3447 158.28 170 10.2652 164.59 10.2806 164.78 10.2954 164.95 10.3102 165.12 10.3248 165.29 10.3394 165.44 175 10.2610 172.26 10.2766 172.36 10.2918 172.45 10.3066 172.53 10.3214 172.62 10.3360 172.70 180 10.2597 180.00 10.2751 180.00 10.2903 180.00 10.3053 180.00 10.3202 180.00 10.3349 180.00 Index Accuracy, of bar-linkage computers, 27, 41 of bar-linkage multipliers, 39 of camoids, 23 of computing mechanisms, 1, 4, 166 of double-ball integrator, 26 of graphical methods for linkage design, versus numerical methods, 199 of squaring cam, 22-23 (See also specific mechanism) Adder, bar-linkage, 36 star linkage in designing of, 256 Additive cells, 6-12 differential belt, 10 bevel-gear, 6-7 cylindrical-gear, 7 with spiral gear, 9 loop-belt, 10-11 screw, 9-10 spur-gear, 7-9 differential worm gearing, 9, 11 Backlash of cylindrical-gear differential, 7 of bar linkages, 27 of plane cams, 19 Backlash error of bar linkages, 21, 33 Ballistic function in vacuum, mechaniza- tion of, 286-299 Bar-linkage adder, accuracy of, 41 approximate, 41, 42 error, 37, 41, 42 structural error of, 41 Bar-linkage computers, compared with integrators, 27 complex, 40-42 concepts of, 43-57 design of, 31-32 terminology of, 43-57 Bar-linkage multipliers, 37-40, 250-283 design of, 37-40, 250-283 error of, 37, 38 39, 40 Bar linkages, 5, 27-42 versus camoids, 23 characteristics of, 32-33 advantages of, 5-6, 32-33 disadvantages of, 33 compared with cams, 33 definition of, 27 dimensional constants of, 31, 32 efficiency of, 28 error of, 27, 28, 31-33 back lash error, 21, 33 elasticity, 33 mechanical, 33 structural, 31-33 frictional losses of, 28 history of use of, 28-31 mechanical features of, 28 straight-line motion by, 29 with one degree of freedom, 34-36 with two degrees of freedom, 37-40, 223-249 adder, 37 multipliers, 37-40, 250-283 as substitute for three-dimensional cam, 37 Belt cam, 20-21 Belt differential, 10 Bevel-gear differential, 6-7 Block diagram in computer design, 3-4 Blokh, Z. Sh., 31n. Camoids versus bar linkages, 23 Cams, 2, 4, 19-23 belt, 20-21 compared with bar linkages, 33, compensated belt, 21-23 cylindrical, 20 definition of, 19 353 354 COMPUTING MECHANISMS AND LINKAGES Cams, linkage, 34 plane, 19 squaring, 21-23 three-dimensional, 23, 37 - Cayley, A., 30n., 31n., 297 Cells, 3, 6-40 additive (see Additive cells) of bar-linkage computers, 41 cams (see Cams) integrators (see Integrators) linear, 6 linkage, 40 multipliers (see Multipliers) resolvers, 15-19 Compensated belt cam, 21-23 Compensating differential, 16 Complementary identification of param- eter and variable, 49 Component solver, 2, 26 Components for computer, selection of, 4 Computational errors, 201 Computer design, 2-5 analytical method of, 3 bar-linkage multipliers, 250-283 components for, 4 constructive method of, 2 error of, 2, 4, 223 selection of components in, 4 function generators with two degrees of freedom, 204-300 harmonic transformer linkages, 58-106 linkage combinations, 166-222 linkages with two degrees of freedom, 223-249 as mechanization of equation, 3 model of, 5 three-bar linkages, 107-165 Computing mechanisms, accuracy of, 1, 4, 166 continuously acting, 1 differential-equation solvers, 1-2 component solvers, 2, 26 integrators, 2 planimeter, 2 speedometers, 2 elementary (see Cells) ideal functional, 43 types of, 1-2 (See also Computers) Constraint linkage in ballistic computer, 297-299 Crank, 27 Crank terminal, 44 Cylindrical cam, 19 Cylindrical-gear differential, backlash of, 7 friction of, 7 Cylindrical rack, 10 D Dawson, G. H., 30n. Design, bar-linkage computer, problem of, 31-32 double-harmonic transformer, 101-106 Differential analyzer, 4 Differential-equation solvers, 2 Differential worm gearing, 9, 11 Differentials, belt, 10 bevel-gear, 6-7 compensating, 16 cylindrical-gear, 7 loop-belt, 10-11 in mechanization of functions, 233, 238 screw, 9-10 with spiral gear, 9 spur-gear, 7-9 Dimensional constants, 32 variations of, 201-202, 205 Direct identification of parameter and variable, 48 Divider, bar-linkage, 40 Division by multipliers, 40 Domain, of parameters, 44 of variable, 44 Double-ball integrators, 24-26 Double harmonic transformers (see Har- monic transformers, double) Double three-bar linkage, 36 in series, 195-198 successive approximations in design of, 196, 197 E Eccentric linkage, 35 as corrective device, 217 Efficiency of bar linkage, 28 Elasticity error of bar linkages, 33 Emch, A., 31n. Error, of bar-linkage adder, 37, 41, 42 of bar-linkage multipliers, 37, 38, 39, 40 of bar linkages, 27, 28, 31, 32, 33 INDEX 355 Error, of computing mechanisms, 2, 4 of double-ball integrators, 26 of linkage multiplier, 4 maximum, 2 of nonideal harmonic transformer, 59, 60, 67-75 of precision squaring devices, 4 of slide multiplier, 4 of squaring cam, 23 (See also specific mechanism) Feedback, 34 Follower (of three-dimensional cam), 23 Friction, of bar linkages, 32 of camoids, 23 of cylindrical-gear differential, 7 Friction-wheel integrators, 24 Function, mechanization of, 46-47 Function generators, 1 bar-linkage computers, 27-42 compare with integrators, 23 with two degrees of freedom, 284-299 factorization of, 168-174 generated by three-bar linkages, 122- 127, 146 grid structure of, 228-233 with ideal grid structure, 233 of two independent parameters, 224 of two independent variables, mechan- ized by bar linkages, 223 mechanized by three-dimensional cams, 23, 223 Gauge of precision of linkage, 201 Gauging constant, 202 Gauging error, 201, 202, 203, 205, 211- 215 Gauging parameters, 200-217 in adjusting linkage constants, 201-205 in eccentric linkage, 218 in three-bar linkage design, 207-217 Geometric method, in mechanization of logarithmic function, 156-165 for three-bar linkage design, 145-165 Graphical methods of linkage design, 199 Greenhill, A. G., 30w. Grid generator, definition of, 225 gauging error of, 281-283 for given function, 225 mechanization of function by, 238 nonideal, 238-242 star, with almost ideal grid structure, 250-284 structural error of, 226 Grid structure, function of, 228-233 generalized, 243 ideal, 229-230, 233-238, 240 star grid generators with, 250-284 nonideal, 230-233, 241, 284-299 regularized, 239-242 transformation of, topological, 232- 233, 239, 244, 246, 249, 251 use of, in linkage design, 243-249 H Harmonic-transformer functions, tables of, 63-67, 301-332 Harmonic transformers, 27, 34, 36, 58- 106 double, 36, 77-106 design of, 95-106 ideal, 58-67 in homogeneous parameters, 62-63 ideal double, 77-95 monotonic functions mechanized by, 78, 82-88 mechanization of function by, 61-63 for monotonic functions, 78, 82-88 nonideal, 58-60, 67-77, 249 design of, 75-77 error of, 59, 60, 67-75 structural error of, 59, 60, 67-75 nonideal double, 95-102 for nonmonotonic function, 89-91 parameters in, 61 in series, 77-106 two, with three-bar linkage, 166-195 use of, 61-62 Hart, H., 30 Hart inversor, 30 Hippisley, R. L., 29n., 31, 297 Homogeneous parameters, 43, 47-49, 62- 63, 68-71 ideal harmonic transformer expressed in, 62-63 Homogeneous variables, 43, 47-49, 62, 64-67, 78, 79, 87, 89, 171 356 COMPUTING MECHANISMS AND LINKAGES Ideal double harmonic transformer (see Harmonic transformer, ideal double) Ideal harmonic transformer (see Har- monic transformer, ideal) Identification, complementary, 49 direct, 48-49 Inertia of bar linkages, 33 Input parameter (see Parameter input) Input scale, 46, 51 Input variable, 45 Integrators, 2, 4, 23-26 compared with function generators, 23 component solver, 26 definition of, 23 double-ball, 24-26 friction-wheel, 24 Intersection nomograms, 40, 251 Inversor, Hart, 30 Peaucellier, 29, 30 Inverted function, 184 mechanization of, 138-139, 143-145, 155, 177 Linkage design, numerical methods in, 167-198 numerical versus graphical methods, 199-200 structural error in, 199 use of grid structure in, 243-249 Linkage inversors, 34 Linkage multiplier constants, adjust- ment of, 277-281 Linkage multipliers, 4, 28-29, 223-283 Linkages, bar, 5 with one degree of freedom, 58-198 star, 238 with two degrees of freedom, 37-40, 223-250 design problem for, 223-226 Logarithmic function, 36 Logarithmic linkage, 238 check of, 209-217 improvement of, by eccentric linkage, 219-222 by gauging-parameter method, 209 217 Loop-belt differential, 10-11 M Johnson, W. W., 31n. K Karpin, E. B., n. Kempe, A. B., 30n. Laverty, W. A., 30n. Least-square method in linkage problems, 206-207 Levenburg, K., 207 Lever, 27 Linear cells (see Additive cells) Linear mechanization, 46, 47, 48 Linear terminal, 46 Link, 27 Linkage cams, 34 Linkage computers, definition of, 44 Linkage constants, final adjustment of, eccentric linkage in, 217-222 gauging parameters in, 200-217 Linkage design, graphical versus numer- ical methods, 199-200 Mathematical design, 4-5 Mechanical error, of bar linkage, 33 Mechanism, regular, 45, 223 Mechanization of alignment nomogram with three parallel straight lines, 37 Mechanization of ballistic function in vacuum, 286-299 Mechanization of a function, 46-47 by combination of three-bar linkage and two ideal harmonic trans- formers, 171, 186 with a discontinuity in derivative, 114- 117 by harmonic transformer, 61-63, 166 by homogeneous parameters and varia- bles, 62-67 by ideal double harmonic transformer, 78-95 with ideal grid structure, 250, 272 by linkage combinations, 166 by method of least squares, 62 by nonideal double harmonic trans- former, 95-101 with one degree of freedom, 166-198 by three-bar linkage, 166, 184 INDEX 357 Mechanization of inverted function by nomographic method, 138-139, 143- 145, 155 Mechanization of logarithm function by double three-bar linkage, 36 by geometric method, 156-166 Mechanization of monotonic functions, 78 Mechanization of relation between varia- bles, 2 Mechanization of tangent function, 171, 175, 177, 178, 180, 182, 183, 186 by ideal harmonic transformer, 165 Model of computer, 5 Monotonic function, mechanization of, 78, 82-88 Multipliers, 12-15 bar-linkage, 37-40, 250-283 definition of, 12 full-range, 40 half-range, 40 linkage, 4 nomographic, 14-15 quarter-range, 40 resolver, 15-19 slide, 4, 12-14 star linkage in designing of, 256-283 transformation of grid structure of, 258-264 with uniform scales, 272 N Nomogram, 13, 14-15, 37, 176, 184 three-bar-linkage, 120-145, 333-352 three-bar-linkage functions represented by, 120-127 for transformer linkage, 275 Nomographic chart, in three-bar-linkage calculations, 120-122 Nomographic method, 178, 181, 183 with three-bar linkage, 174 for three-bar-linkage design, 118-145 Nomographic multipliers, 14-15, 40 Nonideal double harmonic transformer (see Harmonic transformer, non- ideal double) Nonideal harmonic transformer (see Harmonic tiansformer, nonideal) Nonlinear mechanization, 47 Nonmonotonic function, 89-91, 169, 170, 172 Numerical methods of linkage design, 199 Operator, inverse, 50-51 Operator formalism, 49-51 Operator notation, 49-57 Operator symbolism, 185 Operators, 49-57 for double harmonic transformer, 79r 81, 86, 93, 95, 96 equation of, 50, 51, 53 graphical representation of, 51-54 product, 51—53 square of, 54 square-root of, 54-57 Output parameter (see Parameter, out- put) Overlay, construction of, 133-137, 136- 137, 143 example of construction of, 176 use of, 135-136, 141-142, 150, 151, 152, 153, 154 Output scale, 46, 51 Output variable, 45 Parallelogram linkage, 28, 29, 34, 113, 115, 127 Parameters, 6 definition of, 44 domain of, 44-45 of harmonic transformer, 61 homogeneous (see Homogeneous pa- rameters) input, 10, 19, 44, 45 of component solver, 26 of linkages with two degrees of freedom, 223 output, 11, 19, 44, 45 of component solver, 26 of linkages with two degrees of free^ dom, 223 Peaucellier, 29 Peaucellier inversor, 29, 30, 95 Pin gearing, 19 Plane cams, 19 Planimeter, 2 358 COMPUTING MECHANISMS AND LINKAGES Precision, of camoid, 23 of linkage, gauge of, 201 of squaring cam, 23 Precision squaring devices, 2, 4 error of, 4 (See also Cams) R Regular mechanism, 45 Residual error of bar linkages, 32 Resolvers, 15-19 Restricted parameter, 280, 282 Roberts, S., 297 S Scale, definition of, 45-46 input, 46, 51 output, 51 Scale factor, 61 Screw differential, 11 Selection of components of computer, 4 Self-locking, of multipliers, 12 of Peaucellier inversor, 30 of plane cams, 19 Slide multiplier, 4, 12-14 Slide terminal, 44, 166 Speedometers, 2 Spiral gear, differential with, 9 Spur-gear differential, 7-8 Square-root operator, use of, 197-198 Squaring cam, 21-23 Star linkage, 37, 238 in designing of adders, 256 in designing of multipliers, 256 (See also Star grid generator) Star grid generator, 250-299 for ballistic function in vacuum, 286- 299 as multiplier, 256-257, 264 improvement of, 264-271 Straight-line motion, 29 of bar-linkage adder, 37, 41 Structural error, of bar linkages, 32, 33 definition of, 32 of grid generator, 226 of linkages with two degrees of free- dom, 223 in logarithmic linkage, 216, 219, 222, 223 Structural error, of nonideal harmonic transformer, 59, 64, 67-75 reduction of, 199, 207 of star grid generator and transformer linkages, 277-283 Successive approximation, 81, 91-95, 98 in design of function generators, 286-292 in double three-bar linkage design, 196, 197 in terminal harmonic design, numeri- cal method, 187 graphical method, 187 with three-bar linkage components, 174, 177, 180, 184 in three-bar-linkage design, 129, 137- 140, 143-145, 146, 154, 177 Svoboda, A., 198 Tangent linkage, 193-195 Terminals, 43-44 crank, 44 definition of, 43 input, 11, 43 linear, 46 output, 11, 43-44 slide, 44 travel of, 45 Terminal harmonic transformer, design of, 186-192 Three-bar linkages, 35, 36, 107-166 classification of, 108-112 combined with two harmonic trans- formers, 166-195 equations for, 107-108 field of functions of, 118 functions generated by, 122-127 in series, 195—198 special cases, 112-117 Three-bar linkage component, design of, 174-195 geometric method, 145-156 nomographic method, 118-145 problem of, 117-118, 127-128 Three-bar-linkage nomogram, proper- ties of, 333-352 Three-bar motion, 31 Three-dimensional cams, 23 Tolerances, specified, to be met by com- puter, 2 (See also Specific mechanism) INDEX 359 Topological transformation of nomo- gram, 40 Transferred region, 137 Transferred points, 137 Transformer linkages, 225 of ballistic computer, 296-297 design of, 271-277 geometric method, 276 graphical methods, 276 for noncircular scales, 295-297 Transformers, definition of, 225 logarithmic, 238 Travel of a terminal, 45 Two harmonic transformers, properties of, 166 assembly of linkage combination for, 193-195 design of three-bar-linkage component of, 168-186 factorization of function in, 168-174 redesign of terminal, 186-193 Two harmonic transformers, problem of, 166-168 Variables, complementary, 48 definition of, 45 homogeneous {see Homogeneous vari- ables) input, 45 output, 45 range of, 46 W Watt, James, 28 Working model of computer, 5 Worm gearing, 9 differential, 9, 11 Zero position, 6, 22 CATALOGUE OF DOVER BOOKS Catalogue of Dover Books PHYSICS General physics FOUNDATIONS OF PHYSICS, R. B. Lindsay & H. Margenau. Excellent bridge between semi- popular works & technical treatises. A discussion ot methods of physical description, con- struction of theory; valuable tor physicist with elementary calculus who is interested in ideas that give meaning to data, tools of modern physics. Contents include symbolism, math- ematical equations; space & time foundations of mechanics; probability; physics & con- tinua; electron theory; special & general relativity; quantum mechanics; causality. "Thor- ough and yet not overdetailed. Unreservedly recommended," NATURE (London). Unabridged, corrected edition. List of recommended readings. 35 illustrations, xi + 537pp. 5% x 8. S377 Paperbound *2. 7b FUNDAMENTAL FORMULAS OF PHYSICS, ed. by D. H. Menzel. Highly useful, fully inexpensive reference and study text, ranging from simple to highly sophisticated operations. Mathematics integrated into text — each cnapter stands as short textbook ot field represented. Vol. 1: Statistics, Physical Constants, Special Theory of Relativity, Hydrodynamics, Aerodynamics, Boundary Value Problems in Math. Physics; Viscosity, Electromagnetic Theory, etc. Vol. 2: Sound, Acoustics, Geometrical Optics, Electron Optics, High-Energy Phenomena, Magnetism, Biophysics, much more. Index. Total of 800pp. 53/8 x 8. Vol. 1 S595 Paperbound $2.00 Vol. 2 S596 Paperbound $2.00 MATHEMATICAL PHYSICS, 0. H. Menzel. Thorough one-volume treatment of the mathematical techniques vital for classic mechanics, electromagnetic theory, quantum theory, and rela- tivity. Written by the Harvard Professor of Astrophysics for junior, senior, and graduate courses, it gives clear explanations of all those aspects of function theory, vectors, matrices, dyadics, tensors, partial differential equations, etc., necessary tor the understanding of the various physical theories. Electron theory, relativity, and other topics seldom presented appear here in considerable detail. Scores of definitions, conversion factors, dimensional constants, etc. "More detailed than normal for an advanced text . . . excellent set of sec- tions on Dyadics, Matrices, and Tensors," JOURNAL OF THE FRANKLIN INSTITUTE. Index. 193 problems, with answers, x + 412pp. 53/8 x 8. S56 Paperbound $2.00 THE SCIENTIFIC PAPERS OF J. WILLARD GIBBS. All the published papers of America's outstand- ing theoretical scientist (except for "Statistical Mechanics" and "Vector Analysis"). Vol I (thermodynamics) contains one of the most brilliant of all 19th-century scientific papers — the 300-page "On the Equilibrium of Heterogeneous Substances," which founded the science of physical chemistry, and clearly stated a number of highly important natural laws for the first time; 8 other papers complete the first volume. Vol II includes 2 papers on dynamics, 8 on vector analysis and multiple algebra, 5 on the electromagnetic theory of light, and 6 miscella- neous papers. Biographical sketch by H. A. Bumstead. Total of xxxvi + 718pp. 5% x 83/8. S721 Vol I Paperbound $2.50 S722 Vol II Paperbound $2.00 The set $4.50 BASIC THEORIES OF PHYSICS, Peter Gabriel Bergmann. Two-volume set which presents a critical examination of important topics in the major subdivisions of classical and modern physics. The first volume is concerned with classical mechanics and electrodynamics: mechanics of mass points, analytical mechanics, matter in bulk, electrostatics and magneto- statics, electromagnetic interaction, the field waves, special relativity, and waves. The second volume (Heat and Quanta) contains discussions of the kinetic hypothesis, physics and statistics, stationary ensembles, laws of thermodynamics, early quantum theories, atomic spectra, probability waves, quantization in wave mechanics, approximation methods, and abstract quantum theory. A valuable supplement to any thorough course or text. Heat and Quanta: Index. 8 figures, x + 300pp. 5% x 8V2. S968 Paperbound $1.75 Mechanics and Electrodynamics: Index. 14 figures, vii + 280pp. 5% x 8V2. S969 Paperbound $1.75 THEORETICAL PHYSICS, A. S. Kompaneyets. One of the very few thorough studies of the subject in this price range. Provides advanced students with a comprehensive theoretical background. Especially strong on recent experimentation and developments in quantum theory. Contents: Mechanics (Generalized Coordinates, Lagrange's Equation, Collision of Particles, etc.), Electrodynamics (Vector Analysis, Maxwell's equations, Transmission of Signals, Theory of Relativity, etc.), Quantum Mechanics (the Inadequacy of Classical Mechan- ics, the Wave Equation, Motion in a Central Field, Quantum Theory of Radiation, Quantum Theories of Dispersion and Scattering, etc.), and Statistical Physics (Equilibrium Distribution of Molecules in an Ideal Gas, Boltzmann statistics, Bose and Fermi Distribution, Thermodynamic Quantities, etc.). Revised to 1961. Translated by George Yankovsky, author- ized by Kompaneyets. 137 exercises. 56 figures. 529pp. 53/s x 8V2. S972 Paperbound $2.50 ANALYTICAL AND CANONICAL FORMALISM IN PHYSICS, Andre Mercier. A survey, in one vol- ume, of the variational principles (the key principles — in mathematical form — from which the basic laws of any one branch of physics can be derived) of the several branches of physical theory, together with an examination of the relationships among them. Contents: the Lagrangian Formalism, Lagrangian Densities, Canonical Formalism, Canonical Form of Electrodynamics, Hamiltonian Densities, Transformations, and Canonical Form with Vanishing Jacobian Determinant. Numerous examples and exercises. For advanced students, teachers, etc. 6 figures. Index, viii + 222pp. 53/s x 8V2. S1077 Paperbound $1.75 Catalogue of Dover Books HYDRODYNAMICS, H. Dryden, F. Murnaghan, Harry Bateman. Published by the National Research Council in 1932 this enormous volume offers a complete coverage of classical hydrodynamics. Encyclopedic in quality. Partial contents: physics of fluids, motion, turbulent flow, compressible fluids, motion in 1, 2, 3 dimensions; viscous fluids rotating, laminar motion, resistance of motion through viscous fluid, eddy viscosity, hydraulic flow in channels of various shapes, discharge of gases, flow past obstacles, etc. Bibliography of over 2,900 items. Indexes. 23 figures. 634pp. 53/8 x 8. S303 Paperbound $2.75 Mechanics, dynamics, thermodynamics, elasticity MECHANICS, J. P. Den Hartog. Already a classic among introductory texts, the M.I.T. profes- sor's lively and discursive presentation is equally valuable -as a beginner's text, an engineering student's refresher, or a practicing engineer's reference. Emphasis in this highly readable text is on illuminating fundamental principles and showing how they are embodied in a great number of real engineering and design problems: trusses, loaded cables, beams, jacks, hoists, etc. Provides advanced material on relative motion and gyroscopes not usual in introductory texts. "Very thoroughly recommended to all those anxious to improve their real understanding of the principles of mechanics." MECHANICAL WORLD. Index. List of equations. 334 problems, all with answers. Over 550 diagrams and drawings, ix + 462pp. 53/s x 8. • S754 Paperbound $2.00 THEORETICAL MECHANICS: AN INTRODUCTION TO MATHEMATICAL PHYSICS, J. S. Ames, F. 0. Murnaghan. A mathematically rigorous development of theoretical mechanics for the ad- vanced student, with constant practical applications. Used in hundreds of advanced courses. An unusually thorough coverage of gyroscopic and baryscopic material, detailed analyses of the Coriolis acceleration, applications of Lagrange's equations, motion of the double pen- dulum, Hamilton-Jacobi partial differential equations, group velocity and dispersion, etc. Special relativity is also included. 159 problems. 44 figures, ix + 462pp. 53/s x 8. S461 Paperbound $2.25 THEORETICAL MECHANICS: STATICS AND THE DYNAMICS OF A PARTICLE, W. D. MacMillan. Used for over 3 decades as a self-contained and extremely comprehensive advanced under- graduate text in mathematical physics, physics, astronomy, and deeper foundations of engi- neering. Early sections require only a knowledge of geometry; later, a working knowledge of calculus. Hundreds of basic problems, including projectiles to the moon, escape velocity, harmonic motion, ballistics, falling bodies, transmission of power, stress and strain, elasticity, astronomical problems. 340 practice problems plus many fully worked out examples make it possible to test and extend principles developed in the text. 200 figures, xvii + 430pp. 53/s x 8. S467 Paperbound $2.00 THEORETICAL MECHANICS: THE THEORY OF THE POTENTIAL, W. D. MacMillan. A comprehensive, well balanced presentation of potential theory, serving both as an introduction and a refer- ence work with regard to specific problems, for physicists and mathematicians. No prior knowledge of integral relations is assumed, and all mathematical material is developed as it becomes necessary. 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Last work of the great Yale mathematical physicist, still one of the most fundamental treatments available for advanced students and workers in the field. Covers the basic principle of conservation of frobability of phase, theory of errors in the calculated phases of a system, the contribu- ions of Ulausius, Maxwell, Boltzmann, and Gibbs himself, and much more. Includes valuable comparison of statistical mechanics with thermodynamics: Carnot's cycle, mechanical defini- tions of entropy, etc. xvi + 208pp. 53/8 x 8. S707 Paperbound $1.45 Catalogue of Dover Books FOUNDATIONS OF POTENTIAL THEORY, 0. 0. Kellogg. Based on courses given at Harvard this is suitable for both advanced and beginning mathematicians. Proofs are rigorous, and much material not generally available elsewhere is included. Partial contents: forces of gravity, fields of force, divergence theorem, properties of Newtonian potentials at points of free space, potentials as solutions of Laplace's equations, harmonic functions, electrostatics, electric images, logarithmic potential, etc. One of Grundlehren Series, ix + 384pp. 53/e x 8. S144 Paperbound $1.98 THERMODYNAMICS, Enrico Fermi. Unabridged reproduction of 1937 edition. Elementary in treatment; remarkable for clarity, organization. Requires no knowledge of advanced math beyond calculus, only familiarity with fundamentals of thermometry, calorimetry. Partial Contents: Thermodynamic systems; First & Second laws of thermodynamics; Entropy; Thermo- dynamic potentials: phase rule, reversible electric cell; Gaseous reactions: van't Hoff reaction box, principle of LeChatelier; Thermodynamics of dilute solutions: osmotic & vapor pressures, boiling & freezing points; Entropy constant. Index. 25 problems. 24 illustrations, x + 160pp. 53/s x 8 S361 Paperbound $1.75 THE THERMODYNAMICS OF ELECTRICAL PHENOMENA IN METALS and A CONDENSED COLLEC- TION OF THERMODYNAMIC FORMULAS, P. W. Bridgman. Major work by the Nobel Prizewinner: stimulating conceptual introduction to aspects ot the electron theory of metals, giving an intuitive understanding of fundamental relationships concealed by the formal systems of Onsager and others. Elementary mathematical formulations show clearly the fundamental thermodynamica! relationships of the electric field, and a complete phenomenological theory of metals is created. This is the work in which Bridgman announced his famous "thermo- motive force" and his distinction between "driving" and "working" electromotive force. We have added in this Dover edition the author's long unavailable tables of thermo- dynamic formulas, extremely valuable for the speed of reference they allow. Two works bound as one. Index. 33 figures. Bibliography, xviii + 256pp. 53/8 x 8. S723 Paperbound $1.65 TREATISE ON THERMODYNAMICS, Max Planck. Based on Planck's original papers this offers a uniform point of view for the entire field and has been used as an introduction for students who have studied elementary chemistry, physics, and calculus. Rejecting the earlier approaches of Helmholtz and Maxwell, the author makes no assumptions regarding the nature of heat, but begins with a few empirical facts, and from these deduces new physical and chemical laws. 3rd English edition of this standard text by a Nobel laureate, xvi + 297pp. 53/s x 8. S219 Paperbound $1.75 THE MATHEMATICAL THEORY OF ELASTICITY, A. E. H. Love. A wealth of practical illustration combined with thorough discussion of fundamentals — theory, application, special problems and solutions. Partial Contents: Analysis of Strain & Stress, Elasticity of Solid Bodies, Elasticity of Crystals, Vibration of Spheres, Cylinders, Propagation of Waves in Elastic Solid Media, Torsion, Theory of Continuous Beams, Plates. Rigorous treatment of Volterra's theory of dislocations, 2-dimensional elastic systems, other topics of modern interest. "For years the standard treatise on elasticity," AMERICAN MATHEMATICAL MONTHLY. 4th revised edi- tion. Index. 76 figures, xviii + 643pp. 6Va x 9V4. S174 Paperbound $3.00 STRESS WAVES IN SOLIDS, H. Kolsky, Professor of Applied Physics, Brown University. The most readable survey of the theoretical core of current knowledge about the propagation of waves in solids, fully correlated with experimental research. Contents: Part I — Elastic Waves: propagation in an extended plastic medium, propagation in bounded elastic media, experi- mental investigations with elastic materials. Part II — Stress Waves in Imperfectly Elastic Media: internal friction, experimental investigations of dynamic elastic properties, plastic waves and shock waves, fractures produced by stress waves. List of symbols. Appendix. Supplemented bibliography. 3 full-page plates. 46 figures, x + 213pp. 53/8 x 8V2. S1098 Paperbound $1.55 Relativity, quantum theory, atomic and nuclear physics SPACE TIME MATTER, Hermann Weyl. "The standard treatise on the general theory of rela- tivity" (Nature), written by a world-renowned scientist, provides a deep clear discussion of the logical coherence of the general theory, with introduction to all the mathematical tools needed: Maxwell, analytical geometry, non-Euclidean geometry, tensor calculus, etc. Basis is classical space-time, before absorption of relativity. Partial contents: Euclidean space, mathematical form, metrical continuum, relativity of time and space, general theory. 15 dia- grams. Bibliography. New preface for this edition, xviii + 330pp. 53/s x 8. S267 Paperbound $2.00 ATOMIC SPECTRA AND ATOMIC STRUCTURE, G. Herzberg. Excellent general survey for chemists, physicists specializing in other fields. Partial contents: simplest line spectra and elements of atomic theory, building-up principle and periodic system of elements, hyperfine structure of spectral lines, some experiments and applications. Bibliography. 80 figures. Index, xii + 257pp. 53/s x 8. S115 Paperbound $2.00 Catalogue of Dover Books SELECTED PAPERS ON QUANTUM ELECTRODYNAMICS, edited by J. Schwinger. Facsimiles of papers which established quantum electrodynamics, from initial successes through today's position as part of the larger theory of elementary particles. First book publication in any language of these collected papers of Bethe, Bloch, Dirac, Dyson, Fermi, Feynman, Heisen- berg, Kusch, Lamb, Oppenheimer, Pauli, Schwinger, Tomonoga, Weisskopf, Wigner, etc. 34 papers in all, 29 in English, 1 in French, 3 in German, 1 in Italian. Preface and historical commentary by the editor, xvii + 423pp. 6Vb x 9V4. S444 Paperbound $2.75 THE FUNDAMENTAL PRINCIPLES OF QUANTUM MECHANICS, WITH ELEMENTARY APPLICATIONS, E. C. Kemble. An inductive presentation, for the graduate student or specialist in some other branch of physics. Assumes some acquaintance with advanced math; apparatus neces- sary beyond differential equations and advanced calculus is developed as needed. Although a general exposition of principles, hundreds of individual problems are fully treated, with applications of theory being interwoven with development of the mathematical structure. The author is the Professor of Physics at Harvard Univ. ."This excellent book would be of great value to every student ... a rigorous and detailed mathematical discussion of all of the principal quantum-mechanical methods ... has succeeded in keeping his presenta- tions clear and understandable," Dr. Linus Pauling, J. of the American Chemical Society. Appendices: calculus of variations, math, notes, etc. Indexes. 611pp. 5% x 8. S472 Paperbound $3.00 QUANTUM MECHANICS, H. A. Kramers. A superb, up-to-date exposition, covering the most important concepts of quantum theory in exceptionally lucid fashion. 1st half of book shows how the classical mechanics of point particles can be generalized into a consistent quantum mechanics. These 5 chapters constitute a thorough introduction to the foundations of quantum theory. Part II deals with those extensions needed for the application of the theory to problems of atomic and molecular structure. Covers electron spin, the Exclusion Principle, electromagnetic radiation, etc. "This is a book that all who study quantum theory will want to read," J. Polkinghorne, PHYSICS TODAY. Translated by D. ter Haar. Prefaces, introduction. Glossary of symbols. 14 figures. Index, xvi + 496pp. 53/s x 83/s. S1150 Paperbound $2.75 THE THEORY AND THE PROPERTIES OF METALS AND ALLOYS, N. F. Mott, H. Jones. Quantum methods used to develop mathematical models which show interrelationship of basic chem ical phenomena with crystal structure, magnetic susceptibility, electrical, optical properties Examines thermal properties of crystal lattice, electron motion in applied field, cohesion electrical resistance, noble metals, para-, dia-, and ferromagnetism, etc. "Exposition . . clear . . . mathematical treatment . . . simple," Nature. 138 figures. Bibliography. Index xiii + 320pp. 53/s x 8. S456 Paperbound $2.00 FOUNDATIONS OF NUCLEAR PHYSICS, edited by R. T. Beyer. 13 of the most important papers on nuclear physics reproduced in facsimile in the original languages of their authors: the papers most often cited in footnotes, bibliographies. Anderson, Curie, Joliot, Chadwick, Fermi, Lawrence, Cockcroft, Hahn, Yukawa. UNPARALLELED BIBLIOGRAPHY. 122 double- columned pages, over 4,000 articles, books, classified. 57 figures. 288pp. 6Vs x 91/4. S19 Paperbound $2.00 MESON PHYSICS, R. E. Marshak. Traces the basic theory, and explicitly presents results of experiments with particular emphasis on theoretical significance. Phenomena involving mesons as virtual transitions are avoided, eliminating some of the least satisfactory pre- dictions of meson theory. Includes production and study of ir mesons at nonrelativistic nucleon energies, contrasts between n and fi mesons, phenomena associated with nuclear interaction of v mesons, etc. Presents early evidence for new classes of particles and indicates theoretical difficulties created by discovery of heavy mesons and hyperons. Name and subject indices. Unabridged reprint, viii f 378pp. 53/s x 8. S500 Paperbound $1.95 Prices subject to change without notice. Dover publishes books on art, music, philosophy, literature, languages, history, social sciences, psychology, handcrafts, orientalia, puzzles and entertainments, chess, pets and gardens, books explaining science, inter- mediate and higher mathematics, mathematical physics, engineering, biological sciences, earth sciences, classics of science, etc. Write to : Dept. catrr. Dover Publications, Inc. 180 Varick Street, N.Y. Ik, N.Y. (continued from front flap) Applied Mathematics for Radio and Communications Engineers, Carl Smith. $1.75 Fluid Mechanics Through Worked Examples, D. R. L. Smith and /. Houghton. Clothbound $6.00 Mathematical Methods for Scientists and Engineers, L. P. Smith. $2.00 Teach Yourself the Slide Rule, Burns Snodgrass. Clothbound $2.00 An Introduction to the Statistical Dynamics of Control Systems, V. V. Solodovnikov. $2.25 Bridges and Their Builders, David B. Steinman and Sara R. Watson. $2.00 Rayleigh's Principle and Its Applications to Engineering, George Temple and William G. Bickley. $1.50 A History of the Theory of Elasticity and of the Strength of Materials, Isaac Todhunter and Karl Pearson. Clothbound. Three volume set $17.50 Basic Theory and Application of Transistors, U. S. Department of the Army. $1.25 Basic Electricity, U. S. Navy Bureau of Personnel. $3.00 Basic Electronics, U. S. Navy Bureau of Personnel. $2.75 The Schwarz-Christoffel Transformation and Its Applications: A Simple Exposition, Miles Walker. $1.25 Photometry, John W. T. Walsh. $3.00 The Design and Use of Instruments and Accurate Mechanisms: Under- lying Processes, Thomas North Whitehead. $2.00 Teach Yourself Electricity, C. W. Wilman. Clothbound $2.00 Paperbound unless otherwise indicated. Prices subject to change with- out notice. Available at your book dealer or write for free catalogues to Dept. Eng.. Dover Publications, Inc., 180 Varick St., N. Y., N. Y. 10014. Please indicate field of interest. Dover publishes over 125 new books and records each year on such fields as mathematics, physics, explaining science, art, languages, philosophy, classical records, and others. nil", 1 026 2316 5 COMPUTING M EC HAM SMS AND LINKAGES BYANTONIN SVOBODA EDITED BY HUBERT M. JAMES Highly intensified research activities carried on at government laboratories during World War II resulted in major developments in the radio electronics and high-frequency fields. Classified during the war, much of this information was held to be so valuable that it was written up afterwards by a staff of prominent physicists, mathematicians, and engineers at the Radiation Laboratory of M. I. T. The resulting "Radiation Laboratory Series" is recog- nized as the most distinguished and comprehensive series on radio engineer- ing ever published. This present volume is a detailed study of continuously acting computing mechanisms, primarily bar-linkage computers, and the mathematical design for the elements of these mechanisms. Beginning with a brief survey of the standard elements of mechanical computers (additive cells, multipliers, re- solvers, cams, integrators), the author proceeds to a general discussion of bar linkages. The next chapter establishes the terminology and describes the graphical procedures used. Generators of functions of one independent variable are discussed in detail, followed by a thorough treatment of the im- portant mathematical methods in bar-linkage design. The last chapters deal with design methods for bar-linkage generators of functions of two inde- pendent generators. "This volume is, without doubt, an important addition to the literature of computing devices and merits the attention of all engineers and scientists whose activities require familiarity with this specialized field," U. S. Quarterly Booklist. Unabridged and unaltered republication of 1st (1947) edition. Foreword by L. A. DuBridge. Preface by author. 177 figures. 34 tables. 1 foldout. Bibli- ography in footnotes. 2 appendices. Index, xii -(- 359pp. 5% x 8V2- S1404 Paperbound $2.25 A DOVER EDITION DESIGNED FOR YEARS OF USE! We have made every effort to make this the best book possible. 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